Wikiversity:Colloquium

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== [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] ==

I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&amp;#126;2026-28640-56|&amp;#126;2026-28640-56]] ([[User talk:&amp;#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC)

:What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:35, 12 May 2026 (UTC)
: Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC)
::Sounds good -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:13, 15 May 2026 (UTC)
:::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC)

== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)

== [[Wikiversity:Deletion policy]] proposed as policy ==
{{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}}
[[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]].

This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]].

Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy.

=== Voting ===
*{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:33, 18 May 2026 (UTC)
*{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC)

=== Comments ===
{{archive bottom}}

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[[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC)
&lt;!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30513860 --&gt;

== Create an autopatrolled user group? ==

{{tracked|T428269}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)

== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)

== Proposed user group and/or possible policy changes ==

I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)</text>
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        <id>2969951</id>
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      <minor />
      <comment>Changed protection settings for &quot;[[Template:Wikipedia]]&quot;: Disabling cascade protection. ([Edit=Allow only curators and custodians] (indefinite) [Move=Allow only curators and custodians] (indefinite))</comment>
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|D1=Wikipedia|D2=Wikibooks|D3=Wikisource|D4=Wikiquote
|wpd=|wbk=|wsr=|wqt=
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==Usage==
&lt;pre&gt;
  lang=  somelanguage prefix recognized by Wikimedia Foundation servers
 {{{1}}} The article name, defaults to {{PAGENAME}}
&lt;/pre&gt;

Example usage: '''&lt;nowiki&gt;{{Wikipedia | lang=qh | Underwater basket weaving | text=''The Qghlmian Wikipedia article on underwater basketweaving.''}}&lt;/nowiki&gt;'''

{{Wikipedia | lang=qh | Underwater basket weaving | text=''The Qghlmian Wikipedia article on underwater basketweaving.''}}

If only '''{{tl|Wikipedia}}''' is used, the template will default to displaying a link to the English Wikipedia article of the &lt;nowiki&gt;{{PAGENAME}}&lt;/nowiki&gt;.

[[Category:Exclude in print]]
[[Category:Interwiki link templates]]
&lt;!--- Interwiki's ----&gt;

[[fr:Modèle:Wikipédia]]
&lt;/noinclude&gt;</text>
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      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>Dark mode support.</comment>
      <origin>2813357</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="632" sha1="s6wienuw6ij77f2rshn87s307rp9pab" xml:space="preserve">&lt;div class=&quot;projectbox projectbox-border-{{{theme|9}}}&quot;&gt;
&lt;table class=&quot;projectbox-table projectbox-first-background-{{{theme|9}}}&quot; style=&quot;height:{{{height|100%}}};&quot;&gt;
&lt;td class=&quot;projectbox-image projectbox-second-background-{{{theme|9}}}&quot;&gt;[[Image:{{{icon|Crystal kthememgr.svg}}}|36px]]&lt;/td&gt;
&lt;td class=&quot;projectbox-text&quot;&gt;{{{text|This is a project box template}}}&lt;/td&gt;
&lt;/table&gt;
&lt;/div&gt;&lt;templatestyles src=&quot;Projectbox/styles.css&quot; /&gt;&lt;noinclude&gt;{{Documentation}}&lt;/noinclude&gt;&lt;includeonly&gt;{{#ifeq:{{NAMESPACE}}|Template|{{#ifeq:{{{doc|no}}}|yes|{{Projectbox/doc}}}}}} {{#ifeq:{{NAMESPACE}}|Template|[[Category:project boxes]]}}&lt;/includeonly&gt;</text>
      <sha1>s6wienuw6ij77f2rshn87s307rp9pab</sha1>
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      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <minor />
      <comment>Changed protection settings for &quot;[[Template:Projectbox]]&quot;: Disabling cascade protection. ([Edit=Allow only curators and custodians] (indefinite) [Move=Allow only curators and custodians] (indefinite))</comment>
      <origin>2813357</origin>
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      <text bytes="632" sha1="s6wienuw6ij77f2rshn87s307rp9pab" xml:space="preserve">&lt;div class=&quot;projectbox projectbox-border-{{{theme|9}}}&quot;&gt;
&lt;table class=&quot;projectbox-table projectbox-first-background-{{{theme|9}}}&quot; style=&quot;height:{{{height|100%}}};&quot;&gt;
&lt;td class=&quot;projectbox-image projectbox-second-background-{{{theme|9}}}&quot;&gt;[[Image:{{{icon|Crystal kthememgr.svg}}}|36px]]&lt;/td&gt;
&lt;td class=&quot;projectbox-text&quot;&gt;{{{text|This is a project box template}}}&lt;/td&gt;
&lt;/table&gt;
&lt;/div&gt;&lt;templatestyles src=&quot;Projectbox/styles.css&quot; /&gt;&lt;noinclude&gt;{{Documentation}}&lt;/noinclude&gt;&lt;includeonly&gt;{{#ifeq:{{NAMESPACE}}|Template|{{#ifeq:{{{doc|no}}}|yes|{{Projectbox/doc}}}}}} {{#ifeq:{{NAMESPACE}}|Template|[[Category:project boxes]]}}&lt;/includeonly&gt;</text>
      <sha1>s6wienuw6ij77f2rshn87s307rp9pab</sha1>
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  <page>
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        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <minor />
      <comment>Changed protection settings for &quot;[[Template:Minorwiki]]&quot;: Disabling cascade protection. ([Edit=Allow only curators and custodians] (indefinite) [Move=Allow only curators and custodians] (indefinite))</comment>
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&lt;table width=100% cellpadding=0 cellspacing=0&gt;&lt;tr&gt;&lt;td&gt;
* '''''[[wikisource:Special:Search/{{ucfirst:{{{1|{{PAGENAME}}}}}}}|Wikisource]]'''''
* '''''[[wikinews:Special:Search/{{ucfirst:{{{1|{{PAGENAME}}}}}}}|Wikinews]]'''''
* '''''[[wikiquote:Special:Search/{{ucfirst:{{{1|{{PAGENAME}}}}}}}|Wikiquote]]'''''
&lt;/td&gt;&lt;td&gt;
* '''''[[wiktionary:Special:Search/{{ucfirst:{{{1|{{PAGENAME}}}}}}}|Wiktionary]]'''''
* '''''[[wikibooks:Special:Search/{{ucfirst:{{{1|{{PAGENAME}}}}}}}|Wikibooks]]'''''
* '''''[[wikispecies:Special:Search/{{ucfirst:{{{1|{{PAGENAME}}}}}}}|Wikispecies]]'''''
&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
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  <page>
    <title>Template:Projectbox/doc</title>
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      <id>2813358</id>
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      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>+</comment>
      <origin>2813358</origin>
      <model>wikitext</model>
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      <text bytes="5281" sha1="nrcictfyajn6pu11bhw4wfdc1347syq" xml:space="preserve">&lt;noinclude&gt;''This is the documentation for project boxes. It gets transcluded into project box templates. See [[Template:Projectbox]].'' [[Category:project boxes]]&lt;/noinclude&gt;
&lt;div style=&quot;clear:both&quot;&gt;
&lt;/div&gt;
{{TemplateStyles|Template:Projectbox/styles.css}}
===Purpose===

The main purpose of project boxes is '''''organisational'''''.
# Project boxes help categorise educational resources on [[Wikiversity]] into the most important categories by which Wikiversity is organised.
# It's '''''better to categorise by template than by manual category tagging''''' because this allows parser functions to be inserted at a later date which can retrospectively divide 1000's of pages into sub-categories when the main category has got too large. Programmers and scriptwriters can also retrospectively change category names, add extra categories or run other advanced organisational operations on resources as a group. In short: you do a great favour to Wikiversity if major categorisation is by template (project box), not by manual category tagging.
# The words, links, colours and icons are all of secondary importance and can be overridden.
# The words, links, colours and icons are attractive, fun and useful. They serve the purpose of encouraging many users to use project boxes. Of course, not everyone will like them. But many will. Many people who never categorise their resources will be attracted to project boxes because of their appearance. It is unlikely that most people will be attracted by the organisational function - they will place appearance first. But attracting people to project boxes ultimately serves the purpose of better organising Wikiversity.

Why does Wikiversity need this extra layer of organisation?

# Most other Wikimedia projects only have one type of resource per project. Wikiversity has a large and indefinite number of resource types. In other words, Wikiversity content is inherently more complex and therefore needs more organisation.
# Small wikis need low-maintenance solutions. Project boxes are a low-maintenance organisational solution.
# '''Metadata''': this is about sharing catalogues of resources with other educational websites. Promotion of [[open educational resources]] largely occurs through metadata sharing. The project boxes project is largely framed around the common metadata schemes and will allow for future MediaWiki extensions to quickly build metadata catalogues.

=== Usage and customisation ===

You can use all project boxes without any additional configuration at all. Normally they just work by inserting their name into the page code between curly brackets - like this:

{{center|{{tl|{{PAGENAME}}}}}}

However you can do more than this if you wish by specifying additional parameters.

&lt;pre&gt;{{projectbox|theme=[0-14]|icon=[name_of_image]|text=[description]}}&lt;/pre&gt;

====List of parameters ====

''Only valid when this template is fully developed.''

* '''text''': specify an alternative text (overrides the default text).
* '''add''': specify text to append to the end of the default text.
* '''theme''': value 0 to 14; specifies an alternative colour scheme.
* '''icon''': specify an alternative icon; just give it the image name.
* '''style''': currently inactive; will enable the template to adopt different styles.
* '''visible''': currently inactive; will enable a user to toggle the template between categorise-only (invisible) or category+box (visible) modes.

=== Default wording ===

By their nature, project boxes have to be economical with words. The use of few words means that ambiguities, unintended interpretations and other unpleasant things may emerge in the course of time, or the choice of words may unwittingly reflect a particular point of view. Further, many of the original project boxes were created rapidly in large numbers, with little reflection or discussion of the wording. If you wish to suggest alternative default wording, you can use the talk pages for this purpose. It may be better to use the '''''talk page of the corresponding help page''''' for this purpose. When suggesting an alternative default wording:
* consider how many users chose to use the template in its current form;
* consider the option to '''''productively fork''''';
* take into account the option for each user to override the defaults when transcluding;
* consider the option to extend rather than fork a template (because the variables are public);
* consider the need to avoid template clutter.

=== Project box controversy and the deletion of this template ===

Project boxes derive from user boxes. User boxes can be controversial and are often proposed for deletion (e.g. [[w:Wikipedia:Deletion review/Userbox debates/Archived]]). Reasons include problems with default wording (see above) as well as a wider feeling that user boxes are a pointless and distracting fad.

Before proposing this template for deletion:
* please remember that Wikiversity project boxes have a different purpose from user boxes - they serve an essential organisational function as regards the content of Wikiversity;
* as a last resort, an alternative is to render the project box invisible by default - i.e. preserve its categorisation function.

=== List of project box help pages ===
{{Help:project_boxes/list}}
__NOTOC__</text>
      <sha1>nrcictfyajn6pu11bhw4wfdc1347syq</sha1>
    </revision>
    <revision>
      <id>2813364</id>
      <parentid>2813358</parentid>
      <timestamp>2026-06-07T00:30:26Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>Moving the TemplateStyles to noinclude.</comment>
      <origin>2813364</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5304" sha1="icaqbxmkxyzjhx2z0h6ve99iaou7rb2" xml:space="preserve">&lt;noinclude&gt;''This is the documentation for project boxes. It gets transcluded into project box templates. See [[Template:Projectbox]].'' [[Category:project boxes]]&lt;/noinclude&gt;
&lt;div style=&quot;clear:both&quot;&gt;
&lt;/div&gt;
&lt;noinclude&gt;{{TemplateStyles|Template:Projectbox/styles.css}}&lt;/noinclude&gt;
===Purpose===

The main purpose of project boxes is '''''organisational'''''.
# Project boxes help categorise educational resources on [[Wikiversity]] into the most important categories by which Wikiversity is organised.
# It's '''''better to categorise by template than by manual category tagging''''' because this allows parser functions to be inserted at a later date which can retrospectively divide 1000's of pages into sub-categories when the main category has got too large. Programmers and scriptwriters can also retrospectively change category names, add extra categories or run other advanced organisational operations on resources as a group. In short: you do a great favour to Wikiversity if major categorisation is by template (project box), not by manual category tagging.
# The words, links, colours and icons are all of secondary importance and can be overridden.
# The words, links, colours and icons are attractive, fun and useful. They serve the purpose of encouraging many users to use project boxes. Of course, not everyone will like them. But many will. Many people who never categorise their resources will be attracted to project boxes because of their appearance. It is unlikely that most people will be attracted by the organisational function - they will place appearance first. But attracting people to project boxes ultimately serves the purpose of better organising Wikiversity.

Why does Wikiversity need this extra layer of organisation?

# Most other Wikimedia projects only have one type of resource per project. Wikiversity has a large and indefinite number of resource types. In other words, Wikiversity content is inherently more complex and therefore needs more organisation.
# Small wikis need low-maintenance solutions. Project boxes are a low-maintenance organisational solution.
# '''Metadata''': this is about sharing catalogues of resources with other educational websites. Promotion of [[open educational resources]] largely occurs through metadata sharing. The project boxes project is largely framed around the common metadata schemes and will allow for future MediaWiki extensions to quickly build metadata catalogues.

=== Usage and customisation ===

You can use all project boxes without any additional configuration at all. Normally they just work by inserting their name into the page code between curly brackets - like this:

{{center|{{tl|{{PAGENAME}}}}}}

However you can do more than this if you wish by specifying additional parameters.

&lt;pre&gt;{{projectbox|theme=[0-14]|icon=[name_of_image]|text=[description]}}&lt;/pre&gt;

====List of parameters ====

''Only valid when this template is fully developed.''

* '''text''': specify an alternative text (overrides the default text).
* '''add''': specify text to append to the end of the default text.
* '''theme''': value 0 to 14; specifies an alternative colour scheme.
* '''icon''': specify an alternative icon; just give it the image name.
* '''style''': currently inactive; will enable the template to adopt different styles.
* '''visible''': currently inactive; will enable a user to toggle the template between categorise-only (invisible) or category+box (visible) modes.

=== Default wording ===

By their nature, project boxes have to be economical with words. The use of few words means that ambiguities, unintended interpretations and other unpleasant things may emerge in the course of time, or the choice of words may unwittingly reflect a particular point of view. Further, many of the original project boxes were created rapidly in large numbers, with little reflection or discussion of the wording. If you wish to suggest alternative default wording, you can use the talk pages for this purpose. It may be better to use the '''''talk page of the corresponding help page''''' for this purpose. When suggesting an alternative default wording:
* consider how many users chose to use the template in its current form;
* consider the option to '''''productively fork''''';
* take into account the option for each user to override the defaults when transcluding;
* consider the option to extend rather than fork a template (because the variables are public);
* consider the need to avoid template clutter.

=== Project box controversy and the deletion of this template ===

Project boxes derive from user boxes. User boxes can be controversial and are often proposed for deletion (e.g. [[w:Wikipedia:Deletion review/Userbox debates/Archived]]). Reasons include problems with default wording (see above) as well as a wider feeling that user boxes are a pointless and distracting fad.

Before proposing this template for deletion:
* please remember that Wikiversity project boxes have a different purpose from user boxes - they serve an essential organisational function as regards the content of Wikiversity;
* as a last resort, an alternative is to render the project box invisible by default - i.e. preserve its categorisation function.

=== List of project box help pages ===
{{Help:project_boxes/list}}
__NOTOC__</text>
      <sha1>icaqbxmkxyzjhx2z0h6ve99iaou7rb2</sha1>
    </revision>
    <revision>
      <id>2813365</id>
      <parentid>2813364</parentid>
      <timestamp>2026-06-07T00:32:32Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>Modifying.</comment>
      <origin>2813365</origin>
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      <text bytes="5328" sha1="mfoslo1to021l2ew3wkbdadsqele8zk" xml:space="preserve">&lt;noinclude&gt;''This is the documentation for project boxes. It gets transcluded into project box templates. See [[Template:Projectbox]].'' [[Category:project boxes]]&lt;/noinclude&gt;
&lt;div style=&quot;clear:both&quot;&gt;
&lt;/div&gt;
{{#ifeq:{{FULLPAGENAME}}|Template:Projectbox|{{TemplateStyles|Template:Projectbox/styles.css}}}}
===Purpose===

The main purpose of project boxes is '''''organisational'''''.
# Project boxes help categorise educational resources on [[Wikiversity]] into the most important categories by which Wikiversity is organised.
# It's '''''better to categorise by template than by manual category tagging''''' because this allows parser functions to be inserted at a later date which can retrospectively divide 1000's of pages into sub-categories when the main category has got too large. Programmers and scriptwriters can also retrospectively change category names, add extra categories or run other advanced organisational operations on resources as a group. In short: you do a great favour to Wikiversity if major categorisation is by template (project box), not by manual category tagging.
# The words, links, colours and icons are all of secondary importance and can be overridden.
# The words, links, colours and icons are attractive, fun and useful. They serve the purpose of encouraging many users to use project boxes. Of course, not everyone will like them. But many will. Many people who never categorise their resources will be attracted to project boxes because of their appearance. It is unlikely that most people will be attracted by the organisational function - they will place appearance first. But attracting people to project boxes ultimately serves the purpose of better organising Wikiversity.

Why does Wikiversity need this extra layer of organisation?

# Most other Wikimedia projects only have one type of resource per project. Wikiversity has a large and indefinite number of resource types. In other words, Wikiversity content is inherently more complex and therefore needs more organisation.
# Small wikis need low-maintenance solutions. Project boxes are a low-maintenance organisational solution.
# '''Metadata''': this is about sharing catalogues of resources with other educational websites. Promotion of [[open educational resources]] largely occurs through metadata sharing. The project boxes project is largely framed around the common metadata schemes and will allow for future MediaWiki extensions to quickly build metadata catalogues.

=== Usage and customisation ===

You can use all project boxes without any additional configuration at all. Normally they just work by inserting their name into the page code between curly brackets - like this:

{{center|{{tl|{{PAGENAME}}}}}}

However you can do more than this if you wish by specifying additional parameters.

&lt;pre&gt;{{projectbox|theme=[0-14]|icon=[name_of_image]|text=[description]}}&lt;/pre&gt;

====List of parameters ====

''Only valid when this template is fully developed.''

* '''text''': specify an alternative text (overrides the default text).
* '''add''': specify text to append to the end of the default text.
* '''theme''': value 0 to 14; specifies an alternative colour scheme.
* '''icon''': specify an alternative icon; just give it the image name.
* '''style''': currently inactive; will enable the template to adopt different styles.
* '''visible''': currently inactive; will enable a user to toggle the template between categorise-only (invisible) or category+box (visible) modes.

=== Default wording ===

By their nature, project boxes have to be economical with words. The use of few words means that ambiguities, unintended interpretations and other unpleasant things may emerge in the course of time, or the choice of words may unwittingly reflect a particular point of view. Further, many of the original project boxes were created rapidly in large numbers, with little reflection or discussion of the wording. If you wish to suggest alternative default wording, you can use the talk pages for this purpose. It may be better to use the '''''talk page of the corresponding help page''''' for this purpose. When suggesting an alternative default wording:
* consider how many users chose to use the template in its current form;
* consider the option to '''''productively fork''''';
* take into account the option for each user to override the defaults when transcluding;
* consider the option to extend rather than fork a template (because the variables are public);
* consider the need to avoid template clutter.

=== Project box controversy and the deletion of this template ===

Project boxes derive from user boxes. User boxes can be controversial and are often proposed for deletion (e.g. [[w:Wikipedia:Deletion review/Userbox debates/Archived]]). Reasons include problems with default wording (see above) as well as a wider feeling that user boxes are a pointless and distracting fad.

Before proposing this template for deletion:
* please remember that Wikiversity project boxes have a different purpose from user boxes - they serve an essential organisational function as regards the content of Wikiversity;
* as a last resort, an alternative is to render the project box invisible by default - i.e. preserve its categorisation function.

=== List of project box help pages ===
{{Help:project_boxes/list}}
__NOTOC__</text>
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  <page>
    <title>User:Mattski au</title>
    <ns>2</ns>
    <id>117192</id>
    <revision>
      <id>2813302</id>
      <parentid>2541970</parentid>
      <timestamp>2026-06-06T14:34:32Z</timestamp>
      <contributor>
        <username>Mattski au</username>
        <id>311008</id>
      </contributor>
      <comment>Update</comment>
      <origin>2813302</origin>
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      <text bytes="1495" sha1="jpzvl4fc45vrxut0wcx2usohf4nutlr" xml:space="preserve">Matthew resides in his hometown Canberra, Australia, where he operates a cafe and writes on subjects of interest. He holds a Bachelor of Science (psychology) from the University of Canberra and has worked in the hospitality sector for over fifteen years, gaining extensive experience in front of house operations including within hotels, theatres, function/event spaces and cafes. In his spare time he enjoys reading, researching, attending orchestral music and hosting dinner parties. Wikiversity is used as a [[Wikiversity:Personal learning environment|PLE]] alongside his [http://www.mattski.com.au/ blog].

== Current Projects ==

* [[wikibooks:Bookbinding|Bookbinding]]

== Presentations and work undertaken at [[University of Canberra]] ==

* [[Motivation and emotion/Book/Online social networking|Online Social Networking - How does it make people feel]].
* [[User:Mattski au/Creativity and Mindfulness|Creativity and Mindfulness]]
* [[Mindfulness]]

:* [[Mindfulness Introduction Training Session| Group Based Introduction to Mindfulness Training Session]]
:* [[Mindfulness Training/Mindfulness Training Program (six weeks)| Group Based Mindfulness Training Program (Six Week Course)]]

* [[User:Mattski au/Presentations/Introduction to Social Media| Introduction to Social Media]]

==Contact Details ==
Blog: [http://www.mattski.com.au/ www.mattski.com.au]

E-mail: [http://mailto:hello@mattski.com.au/ hello@mattski.com.au]&lt;br&gt;

{{Userboxtop}}
{{MEP2011}}
{{User UC}}
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    <title>User:Marshallsumter</title>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>{{blocked user}}</comment>
      <origin>2813296</origin>
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      <text bytes="48" sha1="46muzhh3xye0a66chkny0t75x0wa93h" xml:space="preserve">{{Blocked user}}
[[nl:Gebruiker:Marshallsumter]]</text>
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  <page>
    <title>Zagatala State National Park</title>
    <ns>0</ns>
    <id>128850</id>
    <revision>
      <id>2813301</id>
      <parentid>2591282</parentid>
      <timestamp>2026-06-06T14:32:40Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>cleanup</comment>
      <origin>2813301</origin>
      <model>wikitext</model>
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      <text bytes="3052" sha1="7elj0q8uria0dmzkuo9t5r9ku43pdx9" xml:space="preserve">{{cleanup|can this be integrated into a [[Wikiversity:Learning projects|project]]?}}
{{center top}}'''Zagatala State Nature Reserve'''{{center bottom}}

* Year of foundation: 1929 
* Area (hectare): 47349 ha 
* Location: Within the territory of Zagatala and Balakan administrative districts, on the southern micro slope in the middle part of the Major Caucasus Ridge. 

Description: Established in administrative areas of [[Zagatala]] and Balakan regions in 1929. Its area totals to 47349 ha. In During the period of its existence, the reserve has belonged to different organization and its area and borders have been changed. The reserve is situated at a height of 650 – 3,646 m above sea level. The reserve territory has a complicated relief due to spurs of the major ridge extending to the south and south-east: Agkemal, Katslar, Rochigel, Pichgel, Khalagel, Ruchug, Mrovdag and others, which are separated from each other by deep river valleys (canyons). Slopes with a steepness of 40-80 m and more occupy an area of more than 450 hectares; slopes with a steepness of 25-40 m prevail. The relief reflects the activity of glacier and other forms of erosion. 

The asymmetry of the river basin is typical: the mountain slopes, deep canyons and valleys are sharply shaped. The Zaqatala reserve was organized with the purpose of protecting and studying the fauna and flora of the southern slopes of the Major Caucasus. The reserve territory is referred to by botanists as the Iberian area of the Caucasus flora province. At the end of the Tertiary period, the forests of this area had a different composition, with a considerable touch of elements of the Hirkan forests, and were much richer than the present ones. Contemporary flora of the reserve has more than a thousand species. 

Such representatives of ancient plants as rhododendron yellow, Laurocerasus officinalis, Caucasian bilberry-bush, Taxus baccata, maple, Polypodiophyta and others are preserved on this territory. The main [[forest]] – forming species of the reserve – Fagus orientalis, as well as Quercus iberica and Corylus colurna are also referred to as ancient plants. The representatives of rare plants: Taxus baccata, apple-tree, ash-tree, birch-tree, cherry-tree, pear-tree and others are observed as well. The fauna of the reserve is rich in species composition. They are: Dagestan aurochs, chamois, red deer, roe, brown bear, fox, badger, Mustela nivalis, pine marten and stone marten, lynx, squirrel and others. 

There are 104 species of birds, including some birds of prey: long-eared owl, golden eagle (Aquilla chrysaetos), Cerchneis tinnunculus, Neophron percnopterus, griffon (Gyps fulvus), bearded vulture (Cypaetus barbatus), black vulture and others. There are some rare, specially protected species of birds: bearded vulture (Gypaetus barbatus), golden eagle (Aquilla chrysaetus), peregrine (Falco peregrinus), Tetraogallus, Accipiter badius, which are registered in the Red Book.

[[Category:Geography of Eurasia]]
[[Category:Azerbaijan]]
[[Category:Animals]]</text>
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  <page>
    <title>User:Atcovi/to do</title>
    <ns>2</ns>
    <id>145726</id>
    <revision>
      <id>2813355</id>
      <parentid>2812772</parentid>
      <timestamp>2026-06-06T22:54:44Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Wikiversity-Related Works */ expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]</comment>
      <origin>2813355</origin>
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      <text bytes="2960" sha1="na1przzolp4ye7czmtbwjanj4qxwa1e" xml:space="preserve">==Atcovi/to do==

=== Current Projects (2026) ===

* [[Intuitive Calculus]]
* [[User:Atcovi/OGM &amp; Suicide/The Paper]] - OGM x SI in high-risk populations according to the IMV model ''[will be moving this off-wiki]''
* [[User:Atcovi/Journey to Clinical PhD]] - figuring this out; current life goal.

* [[WikiJournal Preprints/Mental health in Sri Lanka]] (and later in August: [[User:Atcovi/APA2026 Abstract]])
====Future Endeavors====
* [[WikiJournal Preprints/Suicide amongst refugees in Sweden]] [https://scholar.google.com/scholar?hl=en&amp;as_sdt=0%2C47&amp;as_ylo=2020&amp;as_yhi=2025&amp;q=Suicide+in+Sweden+refugees&amp;btnG=]
* Get [[User:Atcovi/Spring2024]] &amp; [[User:Atcovi/Psychopathology]] into the mainspace. Develop [[Child psychology]] &amp; [[User:Atcovi/PSYC318W]] into a complete course. Merge [[Validity]] into [[User:Atcovi/PSYC318W|PSYC318W]].
* Develop resources related to [[suicidology]] (3 stress response systems? effects of catecholamines on suicidal ideation? neurobiology of suicidal ideation? relation between autobiographical memory and suicide?), expand [[wikipedia:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]] either through [[WikiJournal of Science]] or WP editing.
=====Wikiversity-Related Works=====
* Promote [[Help:Project boxes]], something very useful and unique to Wikiversity. Focus on trying to not only create more project boxes, but to define resource types used in project boxes.
**Ex, what is a [[:Category:Workshops|workshop]]? What differentiates between an [[Help:Essay|essay]] and a [[Help:Paper|paper]]? What differentiates between a [[Template:Notes|notes resource]] (that may be ''derived'' from a homework assignment) and a [[Help:Assignment|homework assignment]] [small note: this page seems to be created by accident and may need a revamp]?
* [[Wikiversity:Original research and scholarly standards]] &amp; improvements/proposals for [[Wikiversity:Original research]] (ex, [[Template:Original research]] should be a mandatory addition to original research on WV + a notice letting readers know that the work is not established science). Develop other pages related to research ethics, including [[Wikiversity:Research]] &amp; [[Wikiversity:Research ethics]].
** [[Wikiversity:Review board]] - should this be Wikiversity 'crats that review original research proposals? 
* [[Wikiversity:Verifiability]] - start heavily scrutinizing pages that don't meet this criteria. 
* [[Wikiversity:Artificial intelligence]] - &quot;substantial&quot;? What defines &quot;substantial&quot;? 
* Expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]. 
{{Archive box|
{{center top}}'''[[User:Atcovi/to do|To do list]]'''{{center bottom}}
----
{{center top}}'''Archives'''{{center bottom}}
*[[User:Atcovi/to do/Current Projects/2023]]
*[[User:Atcovi/to do/Current Projects/January 4, 2022]]
*[[User:Atcovi/to do/Current Projects/September 2017 - January 2018]]
*[[User:Atcovi/to do/Current Projects/2015]]
----
}}
[[Category:Atcovi's Work]]</text>
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    <revision>
      <id>2813366</id>
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      <timestamp>2026-06-07T00:44:00Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Current Projects (2026) */</comment>
      <origin>2813366</origin>
      <model>wikitext</model>
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      <text bytes="2899" sha1="noio1jerdnomn9x82qp4wc7bulqs2b4" xml:space="preserve">==Atcovi/to do==

=== Current Projects (2026) ===

* [[Intuitive Calculus]]
* [[User:Atcovi/OGM &amp; Suicide/The Paper]] - ''[will be moving this off-wiki]''
* [[User:Atcovi/Journey to Clinical PhD]] - figuring this out; current life goal.

* [[WikiJournal Preprints/Mental health in Sri Lanka]] (and later in August: [[User:Atcovi/APA2026 Abstract]])
====Future Endeavors====
* [[WikiJournal Preprints/Suicide amongst refugees in Sweden]] [https://scholar.google.com/scholar?hl=en&amp;as_sdt=0%2C47&amp;as_ylo=2020&amp;as_yhi=2025&amp;q=Suicide+in+Sweden+refugees&amp;btnG=]
* Get [[User:Atcovi/Spring2024]] &amp; [[User:Atcovi/Psychopathology]] into the mainspace. Develop [[Child psychology]] &amp; [[User:Atcovi/PSYC318W]] into a complete course. Merge [[Validity]] into [[User:Atcovi/PSYC318W|PSYC318W]].
* Develop resources related to [[suicidology]] (3 stress response systems? effects of catecholamines on suicidal ideation? neurobiology of suicidal ideation? relation between autobiographical memory and suicide?), expand [[wikipedia:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]] either through [[WikiJournal of Science]] or WP editing.
=====Wikiversity-Related Works=====
* Promote [[Help:Project boxes]], something very useful and unique to Wikiversity. Focus on trying to not only create more project boxes, but to define resource types used in project boxes.
**Ex, what is a [[:Category:Workshops|workshop]]? What differentiates between an [[Help:Essay|essay]] and a [[Help:Paper|paper]]? What differentiates between a [[Template:Notes|notes resource]] (that may be ''derived'' from a homework assignment) and a [[Help:Assignment|homework assignment]] [small note: this page seems to be created by accident and may need a revamp]?
* [[Wikiversity:Original research and scholarly standards]] &amp; improvements/proposals for [[Wikiversity:Original research]] (ex, [[Template:Original research]] should be a mandatory addition to original research on WV + a notice letting readers know that the work is not established science). Develop other pages related to research ethics, including [[Wikiversity:Research]] &amp; [[Wikiversity:Research ethics]].
** [[Wikiversity:Review board]] - should this be Wikiversity 'crats that review original research proposals? 
* [[Wikiversity:Verifiability]] - start heavily scrutinizing pages that don't meet this criteria. 
* [[Wikiversity:Artificial intelligence]] - &quot;substantial&quot;? What defines &quot;substantial&quot;? 
* Expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]. 
{{Archive box|
{{center top}}'''[[User:Atcovi/to do|To do list]]'''{{center bottom}}
----
{{center top}}'''Archives'''{{center bottom}}
*[[User:Atcovi/to do/Current Projects/2023]]
*[[User:Atcovi/to do/Current Projects/January 4, 2022]]
*[[User:Atcovi/to do/Current Projects/September 2017 - January 2018]]
*[[User:Atcovi/to do/Current Projects/2015]]
----
}}
[[Category:Atcovi's Work]]</text>
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  <page>
    <title>Social Victorians/People/Antrim</title>
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      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
      </contributor>
      <origin>2813326</origin>
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      <text bytes="9673" sha1="39xx3bevbjsovua1ejb83vs80m7hqia" xml:space="preserve">== Also Known As ==
*Family name: McDonnell
*Mr. Schomberg M'Donnell
*Mr. M'Donnel
*The Irish Archives Resources spells it MacDonnell.
*Earl of Antrim
**Captain Mark McDonnell, 5th Earl of Antrim (19 July 1855 – 19 December 1869)
**(19 December 1869 – )

== Acquaintances, Friends and Enemies ==

== Organizations ==

=== Schomberg McDonnell ===
*Eton
*Oxford
*Freemasons
*British Army
*Principal Private Secretary to the Prime Minister, Earl Salisbury (1888 – 1902)&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2020-04-07|title=Schomberg Kerr McDonnell|url=https://en.wikipedia.org/w/index.php?title=Schomberg_Kerr_McDonnell&amp;oldid=949531636|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Secretary to the Office of Works (1902–1912), which brought him into contact with the royal family

== Timeline ==
'''1897 July 2, Friday''', Schomberg McDonnell (Mr. Schomberg M'Donnell) attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did, apparently, his brother Alexander McDonnell.

'''1913 February 27''', Schomberg McDonnell and Ethel Henry Davis Harrison married.

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
[[File:Archduke Ferdinand of Austria-Este.png|thumb|alt=Old portrait of a standing and pointing man|Archduke_Ferdinand_of_Austria-Este]]
=== Schomberg McDonnell ===
[[File:Sir-Schomberg-Kerr-McDonnell-as-Duke-Ferdinand-of-Modena.jpg|thumb|left|alt=Black-and-white photograph of a standing man leaning on the base of a column and richly dressed in an historical costume|Sir Schomberg Kerr McDonnell in costume as Duke Ferdinand of Modena. ©National Portrait Gallery, London.]]
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Schomberg McDonnell (at 104) was dressed as Duke Ferdinand of Modena:
*He was wearing a &quot;Uniform of the period of Marie Thérèse, white cloth, trimmed with gold.&quot;&lt;ref name=&quot;:1&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|p. 8, Col. 1c}}
*Mr. M'Donnel [sic] was dressed as Duke Ferdinand of Modena in the Austrian Court of Maria Theresa Quadrille.&lt;ref name=&quot;:1&quot; /&gt;&lt;ref&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;
*&quot;Mr. Schomberg Macdonnell (period of Marie Thérèse), white cloth uniform trimmed with gold.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|p. 36, Col. 3b}}
Alexander Bassano's portrait of &quot;Sir Schomberg Kerr McDonnell as Duke Ferdinand of Modena&quot; in costume is photogravure #69 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:2&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;The Hon. Schomberg McDonnell as Duke Ferdinand of Modena.&quot;&lt;ref&gt;&quot;Sir Schomberg Kerr McDonnell as Duke Ferdinand of Modena.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158423/Sir-Schomberg-Kerr-McDonnell-as-Duke-Ferdinand-of-Modena.&lt;/ref&gt;

Perhaps by Duke Ferdinand of Moderna, Schomberg McDonnell meant Archduke Ferdinand Karl of Austria-Este (1 June 1754 – 24 December 1806), son of Marie-Thérèse.&lt;ref&gt;{{Cite journal|date=2021-06-02|title=Ferdinand Karl, Archduke of Austria-Este|url=https://en.wikipedia.org/w/index.php?title=Ferdinand_Karl,_Archduke_of_Austria-Este&amp;oldid=1026464956|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Ferdinand_Karl,_Archduke_of_Austria-Este.&lt;/ref&gt; He did not reign as Duke of Moderna, although he had been designated as the Duke, because Napoleon Bonaparte occupied and renamed what had been (and would go back to being) Moderna.&lt;ref&gt;{{Cite journal|date=2021-11-07|title=Duchy of Modena and Reggio|url=https://en.wikipedia.org/w/index.php?title=Duchy_of_Modena_and_Reggio&amp;oldid=1053993037|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Duchy_of_Modena_and_Reggio.&lt;/ref&gt;

The portrait of Archduke Ferdinand of Austria-Este (right) is undated, but it seems to be from later in his life.
[[File:Hon-Alexander-McDonnell-as-Mercutio.jpg|thumb|alt=Black-and-white photograph of a standing man with a mustache richly dressed in an historical costume|Hon. Alexander McDonnell in costume as Mercutio. ©National Portrait Gallery, London.]]
=== Hon. Alexander McDonnell ===
Hon. Alexander McDonnell (at 676) also attended.

Lafayette's portrait of &quot;Hon. Alexander McDonnell as Mercutio&quot; in costume is photogravure #192 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:2&quot; /&gt; The printing on the portrait says, &quot;The Hon. A. McDonnell as Mercutio.&quot;&lt;ref&gt;&quot;A. McDonnell as Mercutio.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158555/Hon-Alexander-McDonnell-as-Mercutio.&lt;/ref&gt;

The ''Gentlewoman'' describes the costume of &quot;Mr. A. Macdonald&quot; as

* &quot;(Romeo), white satin tunic elaborately embroidered in gold and turquoise; turquoise velvet cloak lined with grey.&quot;&lt;ref name=&quot;:3&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 34, Col. 2a}}

The costume in the photograph taken by Lafayette for the Album aligns with the description from the ''Gentlewoman'', given that we can't tell what color the cloak and its lining are. Similarly, this conflation of Romeo and Mercutio seems like a very small mistake in one of these newspaper accounts. For now, then, I'm guessing that this description should be applied to Alexander McDonnell rather than Mcdonald.

A character from William Shakespeare's ''Romeo and Juliet'', a friend of Romeo, Mercutio talks Romeo into attending the Capulet's masquerade ball, which is where he meets Juliet. Mercutio dies in a duel Romeo will not take up, saying the famous line &quot;A plague o' both your houses!&quot;

Henry Irving's Lyceum Theatre seems to have run performances of ''Romeo and Juliet'' in the 1890s, starring Mrs. Patrick Campbell as Juliet&lt;ref&gt;{{Cite journal|date=2021-11-12|title=Mrs Patrick Campbell|url=https://en.wikipedia.org/w/index.php?title=Mrs_Patrick_Campbell&amp;oldid=1054853007|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Mrs_Patrick_Campbell.&lt;/ref&gt; and Johnston Forbes-Robertson taking over as Romeo, for s more realistic, less pictorial interpretation.&lt;ref&gt;{{Cite journal|date=2021-12-13|title=Romeo and Juliet|url=https://en.wikipedia.org/w/index.php?title=Romeo_and_Juliet&amp;oldid=1060140047|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Romeo_and_Juliet.&lt;/ref&gt; In these performances in fall 1895, Mercutio was played by Charles Francis Coghlan (11 June 1842 – 27 November 1899).&lt;ref&gt;{{Cite journal|date=2021-09-11|title=Charles Francis Coghlan|url=https://en.wikipedia.org/w/index.php?title=Charles_Francis_Coghlan&amp;oldid=1043620554|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Charles_Francis_Coghlan.&lt;/ref&gt;

== Demographics ==
*Nationality: Earls of Antrim: Anglo-Irish&lt;ref name=&quot;:0&quot; /&gt;

== Family ==
*Captain Mark McDonnell, 5th Earl of Antrim (3 April 1814 – 19 December 1869)&lt;ref&gt;&quot;Captain Mark McDonnell, 5th Earl of Antrim.&quot; {{Cite web|url=https://www.thepeerage.com/p1245.htm#i12447|title=Person Page|website=www.thepeerage.com|access-date=2020-10-21}}&lt;/ref&gt;
*Jane Emma Hannah Macan (c. 1825 – 21 April 1892)&lt;ref&gt;&quot;Jane Emma Hannah Macan.&quot; {{Cite web|url=https://www.thepeerage.com/p2572.htm#i25711|title=Person Page|website=www.thepeerage.com|access-date=2020-10-21}}&lt;/ref&gt;
#William Randal McDonnell, 6th Earl of Antrim (8 January 1851 – 19 July 1918)
#Hon. Mark Henry Horace McDonnell (18 August 1852 – 23 April 1909)
#Lady Caroline Elizabeth McDonnell (c 1854 – 23 February 1930)
#Hon. Hugh Seymour McDonnell (18 July 1855 – 24 October 1879)
#Hon. '''Alexander McDonnell''' (23 June 1857 – 9 December 1945)
#Lady Mabel Harriet McDonnell (1858 – 31 December 1942)
#Lady Evelyn McDonnell (1860 – 1 June 1947)
#Major Hon. Sir '''Schomberg Kerr McDonnell''' (22 March 1861 – 23 November 1915)
#Lady Jane Grey McDonnell (15 June 1863 – 27 August 1953)
#Lady Helena McDonnell (1865 – 11 January 1948)


*Schomberg Kerr McDonnell (22 March 1861 – 23 November 1915)&lt;ref&gt;&quot;Major Hon. Sir Schomberg Kerr McDonnell.&quot; {{Cite web|url=https://www.thepeerage.com/p866.htm#i8660|title=Person Page|website=www.thepeerage.com|access-date=2020-10-21}}&lt;/ref&gt;
*Ethel Henry Davis Harrison McDonnell ( – 14 April 1916)&lt;ref&gt;&quot;Ethel Henry Davis.&quot; {{Cite web|url=https://www.thepeerage.com/p806.htm#i8056|title=Person Page|website=www.thepeerage.com|access-date=2020-10-21}}&lt;/ref&gt;

== Notes and Questions ==
#Schomberg Kerr McDonnell was the 5th son and 8th child of Mark McDonnell, the 5th Earl of Antrim; his eldest brother was William Randal McDonnell, 6th Earl of Antrim.
#Mr. A. McDonald: No other mention of a Mr. A. McDonald (by any spelling) attending this party exists, which is suggestive rather than conclusive.
#[https://iar.ie/archive/earl-antrim-estate-papers/ Estate papers of the Earls of Antrim] are in the Public Records Office of Northern Ireland. I don't see personal papers listed.

== Footnotes ==
{{reflist}}</text>
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.projectbox-top-4 {
    background-color: #FFE4E1; /* MistyRose */
    border: 1px solid #CD5C5C; /* IndianRed */
}

.projectbox-top-5 {
    background-color: #FFF0F5; /* LavenderBlush */
    border: 1px solid #9370DB; /* MediumPurple */
}

.projectbox-top-6 {
    background-color: #FFFFF0; /* Ivory */
    border: 1px solid #FFD700; /* Gold */
}

.projectbox-top-7 {
    background-color: #F0FFFF; /* Azure */
    border: 1px solid #4169E1; /* RoyalBlue */
}

.projectbox-top-8 {
    background-color: #FFF5EE; /* Seashell */
    border: 1px solid #E78A69;
}

.projectbox-top-9 {
    background-color: #F6FFF1;
    border: 1px solid #71BE3F;
}

.projectbox-top-10 {
    background-color: #F4FAFF;
    border: 1px solid #4290BC;
}

.projectbox-top-11 {
    background-color: #FFF8FF;
    border: 1px solid #C289C3;
}

.projectbox-top-12 {
    background-color: #FFF6F8;
    border: 1px solid #C56B74;
}

.projectbox-top-13 {
    background-color: #F5F5FF;
    border: 1px solid #8488DC;
}

.projectbox-top-14 {
    background-color: #FFFFFF;
    border: 1px solid #AAAAAA;
}

.projectbox-top-15 {
    background-color: #D8BFD8; /* Thistle */
    border: 1px solid #D8BFD8; /* Thistle */
}

.projectbox-1 {
    background-color: #F5DEB3; /* Wheat */
}

.projectbox-2 {
    background-color: #90EE90; /* LightGreen */
}

.projectbox-3 {
    background-color: #B0C4DE; /* LightSteelBlue */
}

.projectbox-4 {
    background-color: #F08080; /* LightCoral */
}

.projectbox-5 {
    background-color: #D8BFD8; /* Thistle */
}

.projectbox-6 {
    background-color: #FFFF99; /* #ff9 shorthand → #FFFF99 */
}

.projectbox-7 {
    background-color: #87CEFA; /* LightSkyBlue */
}

.projectbox-8 {
    background-color: #FFDAB9; /* Peachpuff */
}

.projectbox-9 {
    background-color: #C0EAA6;
}

.projectbox-10 {
    background-color: #9AD4F6;
}

.projectbox-11 {
    background-color: #E6C6E6;
}

.projectbox-12 {
    background-color: #F4B8BF;
}

.projectbox-13 {
    background-color: #CED1FA;
}

.projectbox-14 {
    background-color: #E4E4E4;
}

.projectbox-15 {
    background-color: #D8BFD8; /* Thistle */
}

/* ===========================
   Dark Mode – projectbox-top-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-1 {
		background-color: #2A2620;
		border: 1px solid #B97A58;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-1 {
		background-color: #2A2620;
		border: 1px solid #B97A58;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-2 {
		background-color: #102A20;
		border: 1px solid #1A6620;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-2 {
		background-color: #102A20;
		border: 1px solid #1A6620;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-3 {
		background-color: #12202A;
		border: 1px solid #3A628A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-3 {
		background-color: #12202A;
		border: 1px solid #3A628A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-4 {
		background-color: #2A1A1C;
		border: 1px solid #A24A4A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-4 {
		background-color: #2A1A1C;
		border: 1px solid #A24A4A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-5 {
		background-color: #281A28;
		border: 1px solid #7A52A8;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-5 {
		background-color: #281A28;
		border: 1px solid #7A52A8;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-6 {
		background-color: #2A2818;
		border: 1px solid #CCAA33;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-6 {
		background-color: #2A2818;
		border: 1px solid #CCAA33;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-7 {
		background-color: #112028;
		border: 1px solid #3459A5;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-7 {
		background-color: #112028;
		border: 1px solid #3459A5;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-8 {
		background-color: #2A201A;
		border: 1px solid #C07052;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-8 {
		background-color: #2A201A;
		border: 1px solid #C07052;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-9 {
		background-color: #1B2715;
		border: 1px solid #5BA535;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-9 {
		background-color: #1B2715;
		border: 1px solid #5BA535;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-10 {
		background-color: #14222A;
		border: 1px solid #3578A0;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-10 {
		background-color: #14222A;
		border: 1px solid #3578A0;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-11 {
		background-color: #241A24;
		border: 1px solid #9E6AA9;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-11 {
		background-color: #241A24;
		border: 1px solid #9E6AA9;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-12 {
		background-color: #2A1C1E;
		border: 1px solid #A0525C;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-12 {
		background-color: #2A1C1E;
		border: 1px solid #A0525C;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-13 {
		background-color: #20202A;
		border: 1px solid #6A6EC7;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-13 {
		background-color: #20202A;
		border: 1px solid #6A6EC7;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-14 {
		background-color: #1A1A1A;
		border: 1px solid #777;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-14 {
		background-color: #1A1A1A;
		border: 1px solid #777;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-top-15 {
		background-color: #3A2D3A;
		border: 1px solid #B49BB4;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-top-15 {
		background-color: #3A2D3A;
		border: 1px solid #B49BB4;
	}
}

/* ===========================
   Dark Mode – projectbox-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-1 {
		background-color: #3A2D1A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-1 {
		background-color: #3A2D1A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-2 {
		background-color: #1E3A1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-2 {
		background-color: #1E3A1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-3 {
		background-color: #2A3444;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-3 {
		background-color: #2A3444;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-4 {
		background-color: #3A1E1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-4 {
		background-color: #3A1E1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-5 {
		background-color: #3A2D3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-5 {
		background-color: #3A2D3A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-6 {
		background-color: #4A4A19
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-6 {
		background-color: #4A4A19;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-7 {
		background-color: #1E2F44;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-7 {
		background-color: #1E2F44;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-8 {
		background-color: #3A2918;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-8 {
		background-color: #3A2918;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-9 {
		background-color: #293911;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-9 {
		background-color: #293911;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-10 {
		background-color: #1C2F3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-10 {
		background-color: #1C2F3A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-11 {
		background-color: #342434;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-11 {
		background-color: #342434;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-12 {
		background-color: #40252A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-12 {
		background-color: #40252A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-13 {
		background-color: #26283F;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-13 {
		background-color: #26283F;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-14 {
		background-color: #2A2A2A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-14 {
		background-color: #2A2A2A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-15 {
		background-color: #3A2D3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-15 {
		background-color: #3A2D3A;
	}
}</text>
      <sha1>qohoaomrvnkk4rfvq4jsowhzivk48ct</sha1>
    </revision>
    <revision>
      <id>2813356</id>
      <parentid>2813307</parentid>
      <timestamp>2026-06-07T00:20:28Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>Dark mode support.</comment>
      <origin>2813356</origin>
      <model>sanitized-css</model>
      <format>text/css</format>
      <text bytes="16088" sha1="6hjljwdip8uicwwp5kivfiu1ghu48t7" xml:space="preserve">.projectbox {
	float: right;
	clear: right;
	margin: 5px 0 5px 5px;
	border-width: 1px;
	border-style: solid;
}
.projectbox-table {
	border: 0;
	border-collapse: collapse;
	border-spacing: 0;
	margin: 0;
	width: 228px;
}
.projectbox-image {
	border: 0;
	text-align: center;
	width: 45px;
	height: 45px;
}
.projectbox-text {
	border: 0;
	font-size: 8pt;
	line-height: 1.25em;
	padding: 0.5em;
}

/**
 * Mobile and small screens
 */
@media screen and ( max-width: 720px ) {
	.projectbox {
		margin: 5px 0;
	}
}

.projectbox-border-1 {
	border: 1px solid #E39C79;
}

.projectbox-border-2 {
	border: 1px solid #228B22; /* ForestGreen */
}

.projectbox-border-3 {
    border: 1px solid #4682B4; /* SteelBlue */
}

.projectbox-border-4 {
	border: 1px solid #CD5C5C; /* IndianRed */
}

.projectbox-border-5 {
	border: 1px solid #9370DB; /* MediumPurple */
}

.projectbox-border-6 {
    border: 1px solid #FFD700; /* Gold */
}

.projectbox-border-7 {
    border: 1px solid #4169E1; /* RoyalBlue */
}

.projectbox-border-8 {
    border: 1px solid #E78A69;
}

.projectbox-border-9 {
    border: 1px solid #71BE3F;
}

.projectbox-border-10 {
    border: 1px solid #4290BC;
}

.projectbox-border-11 {
    border: 1px solid #C289C3;
}

.projectbox-border-12 {
    border: 1px solid #C56B74;
}

.projectbox-border-13 {
    border: 1px solid #8488DC;
}

.projectbox-border-14 {
    border: 1px solid #AAAAAA;
}

.projectbox-border-15 {
    border: 1px solid #D8BFD8; /* Thistle */
}

.projectbox-first-background-1 {
    background-color: #FFFCF1;
}

.projectbox-first-background-2 {
    background-color: #F5FFFA; /* MintCream */
}

.projectbox-first-background-3 {
    background-color: #F0F8FF; /* AliceBlue */
}

.projectbox-first-background-4 {
    background-color: #FFE4E1; /* MistyRose */
}

.projectbox-first-background-5 {
    background-color: #FFF0F5; /* LavenderBlush */
}

.projectbox-first-background-6 {
	background-color: #FFFFF0; /* Ivory */
}

.projectbox-first-background-7 {
	background-color: #F0FFFF; /* Azure */
}

.projectbox-first-background-8 {
	background-color: #FFF5EE; /* Seashell */
}

.projectbox-first-background-9 {
	background-color: #F6FFF1;
}

.projectbox-first-background-10 {
	background-color: #F4FAFF;
}

.projectbox-first-background-11 {
	background-color: #FFF8FF;
}

.projectbox-first-background-12 {
    background-color: #FFF6F8;
}

.projectbox-first-background-13 {
	background-color: #F5F5FF;
}

.projectbox-first-background-14 {
	background-color: #FFFFFF;
}

.projectbox-first-background-15 {
	background-color: #D8BFD8; /* Thistle */
}

.projectbox-second-background-1 {
    background-color: #F5DEB3; /* Wheat */
}

.projectbox-second-background-2 {
    background-color: #90EE90; /* LightGreen */
}

.projectbox-second-background-3 {
    background-color: #B0C4DE; /* LightSteelBlue */
}

.projectbox-second-background-4 {
    background-color: #F08080; /* LightCoral */
}

.projectbox-second-background-5 {
    background-color: #D8BFD8; /* Thistle */
}

.projectbox-second-background-6 {
    background-color: #FFFF99; /* #ff9 shorthand → #FFFF99 */
}

.projectbox-second-background-7 {
    background-color: #87CEFA; /* LightSkyBlue */
}

.projectbox-second-background-8 {
    background-color: #FFDAB9; /* Peachpuff */
}

.projectbox-second-background-9 {
    background-color: #C0EAA6;
}

.projectbox-second-background-10 {
    background-color: #9AD4F6;
}

.projectbox-second-background-11 {
    background-color: #E6C6E6;
}

.projectbox-second-background-12 {
    background-color: #F4B8BF;
}

.projectbox-second-background-13 {
    background-color: #CED1FA;
}

.projectbox-second-background-14 {
    background-color: #E4E4E4;
}

.projectbox-second-background-15 {
    background-color: #D8BFD8; /* Thistle */
}

/* ===========================
   Dark Mode – projectbox-border-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-1 {
		border: 1px solid #B97A58;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-1 {
		border: 1px solid #B97A58;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-2 {
		border: 1px solid #1A6620;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-2 {
		border: 1px solid #1A6620;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-3 {
		border: 1px solid #3A628A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-3 {
		border: 1px solid #3A628A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-4 {
		border: 1px solid #A24A4A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-4 {
		border: 1px solid #A24A4A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-5 {
		border: 1px solid #7A52A8;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-5 {
		border: 1px solid #7A52A8;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-6 {
		border: 1px solid #CCAA33;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-6 {
		border: 1px solid #CCAA33;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-7 {
		border: 1px solid #3459A5;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-7 {
		border: 1px solid #3459A5;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-8 {
		border: 1px solid #C07052;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-8 {
		border: 1px solid #C07052;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-9 {
		border: 1px solid #5BA535;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-9 {
		border: 1px solid #5BA535;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-10 {
		border: 1px solid #3578A0;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-10 {
		border: 1px solid #3578A0;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-11 {
		border: 1px solid #9E6AA9;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-11 {
		border: 1px solid #9E6AA9;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-12 {
		border: 1px solid #A0525C;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-12 {
		border: 1px solid #A0525C;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-13 {
		border: 1px solid #6A6EC7;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-13 {
		border: 1px solid #6A6EC7;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-14 {
		border: 1px solid #777;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-14 {
		border: 1px solid #777;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-15 {
		border: 1px solid #B49BB4;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-15 {
		border: 1px solid #B49BB4;
	}
}

/* ===========================
   Dark Mode – projectbox-first-background-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-1 {
		background-color: #2A2620;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-1 {
		background-color: #2A2620;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-2 {
		background-color: #102A20;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-2 {
		background-color: #102A20;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-3 {
		background-color: #12202A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-3 {
		background-color: #12202A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-4 {
		background-color: #2A1A1C;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-4 {
		background-color: #2A1A1C;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-5 {
		background-color: #281A28;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-5 {
		background-color: #281A28;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-6 {
		background-color: #2A2818;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-6 {
		background-color: #2A2818;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-7 {
		background-color: #112028;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-7 {
		background-color: #112028;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-8 {
		background-color: #2A201A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-8 {
		background-color: #2A201A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-9 {
		background-color: #1B2715;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-9 {
		background-color: #1B2715;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-10 {
		background-color: #14222A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-10 {
		background-color: #14222A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-11 {
		background-color: #241A24;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-11 {
		background-color: #241A24;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-13 {
		background-color: #20202A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-13 {
		background-color: #20202A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-14 {
		background-color: #1A1A1A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-14 {
		background-color: #1A1A1A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-15 {
		background-color: #3A2D3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-15 {
		background-color: #3A2D3A;
	}
}

/* ===========================
   Dark Mode – projectbox-second-background-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-1 {
		background-color: #3A2D1A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-1 {
		background-color: #3A2D1A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-2 {
		background-color: #1E3A1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-2 {
		background-color: #1E3A1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-3 {
		background-color: #2A3444;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-3 {
		background-color: #2A3444;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-4 {
		background-color: #3A1E1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-4 {
		background-color: #3A1E1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-5 {
		background-color: #3A2D3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-5 {
		background-color: #3A2D3A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-6 {
		background-color: #4A4A19
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-6 {
		background-color: #4A4A19;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-7 {
		background-color: #1E2F44;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-7 {
		background-color: #1E2F44;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-8 {
		background-color: #3A2918;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-8 {
		background-color: #3A2918;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-9 {
		background-color: #293911;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-9 {
		background-color: #293911;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-10 {
		background-color: #1C2F3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-10 {
		background-color: #1C2F3A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-11 {
		background-color: #342434;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-11 {
		background-color: #342434;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-12 {
		background-color: #40252A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-12 {
		background-color: #40252A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-13 {
		background-color: #26283F;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-13 {
		background-color: #26283F;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-14 {
		background-color: #2A2A2A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-14 {
		background-color: #2A2A2A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-15 {
		background-color: #3A2D3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-15 {
		background-color: #3A2D3A;
	}
}</text>
      <sha1>6hjljwdip8uicwwp5kivfiu1ghu48t7</sha1>
    </revision>
    <revision>
      <id>2813359</id>
      <parentid>2813356</parentid>
      <timestamp>2026-06-07T00:23:41Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>+</comment>
      <origin>2813359</origin>
      <model>sanitized-css</model>
      <format>text/css</format>
      <text bytes="16110" sha1="7fx75ogm7hd0ttoty0sbbl4jbhoppoc" xml:space="preserve">/* {{pp-template}} */
.projectbox {
	float: right;
	clear: right;
	margin: 5px 0 5px 5px;
	border-width: 1px;
	border-style: solid;
}
.projectbox-table {
	border: 0;
	border-collapse: collapse;
	border-spacing: 0;
	margin: 0;
	width: 228px;
}
.projectbox-image {
	border: 0;
	text-align: center;
	width: 45px;
	height: 45px;
}
.projectbox-text {
	border: 0;
	font-size: 8pt;
	line-height: 1.25em;
	padding: 0.5em;
}

/**
 * Mobile and small screens
 */
@media screen and ( max-width: 720px ) {
	.projectbox {
		margin: 5px 0;
	}
}

.projectbox-border-1 {
	border: 1px solid #E39C79;
}

.projectbox-border-2 {
	border: 1px solid #228B22; /* ForestGreen */
}

.projectbox-border-3 {
    border: 1px solid #4682B4; /* SteelBlue */
}

.projectbox-border-4 {
	border: 1px solid #CD5C5C; /* IndianRed */
}

.projectbox-border-5 {
	border: 1px solid #9370DB; /* MediumPurple */
}

.projectbox-border-6 {
    border: 1px solid #FFD700; /* Gold */
}

.projectbox-border-7 {
    border: 1px solid #4169E1; /* RoyalBlue */
}

.projectbox-border-8 {
    border: 1px solid #E78A69;
}

.projectbox-border-9 {
    border: 1px solid #71BE3F;
}

.projectbox-border-10 {
    border: 1px solid #4290BC;
}

.projectbox-border-11 {
    border: 1px solid #C289C3;
}

.projectbox-border-12 {
    border: 1px solid #C56B74;
}

.projectbox-border-13 {
    border: 1px solid #8488DC;
}

.projectbox-border-14 {
    border: 1px solid #AAAAAA;
}

.projectbox-border-15 {
    border: 1px solid #D8BFD8; /* Thistle */
}

.projectbox-first-background-1 {
    background-color: #FFFCF1;
}

.projectbox-first-background-2 {
    background-color: #F5FFFA; /* MintCream */
}

.projectbox-first-background-3 {
    background-color: #F0F8FF; /* AliceBlue */
}

.projectbox-first-background-4 {
    background-color: #FFE4E1; /* MistyRose */
}

.projectbox-first-background-5 {
    background-color: #FFF0F5; /* LavenderBlush */
}

.projectbox-first-background-6 {
	background-color: #FFFFF0; /* Ivory */
}

.projectbox-first-background-7 {
	background-color: #F0FFFF; /* Azure */
}

.projectbox-first-background-8 {
	background-color: #FFF5EE; /* Seashell */
}

.projectbox-first-background-9 {
	background-color: #F6FFF1;
}

.projectbox-first-background-10 {
	background-color: #F4FAFF;
}

.projectbox-first-background-11 {
	background-color: #FFF8FF;
}

.projectbox-first-background-12 {
    background-color: #FFF6F8;
}

.projectbox-first-background-13 {
	background-color: #F5F5FF;
}

.projectbox-first-background-14 {
	background-color: #FFFFFF;
}

.projectbox-first-background-15 {
	background-color: #D8BFD8; /* Thistle */
}

.projectbox-second-background-1 {
    background-color: #F5DEB3; /* Wheat */
}

.projectbox-second-background-2 {
    background-color: #90EE90; /* LightGreen */
}

.projectbox-second-background-3 {
    background-color: #B0C4DE; /* LightSteelBlue */
}

.projectbox-second-background-4 {
    background-color: #F08080; /* LightCoral */
}

.projectbox-second-background-5 {
    background-color: #D8BFD8; /* Thistle */
}

.projectbox-second-background-6 {
    background-color: #FFFF99; /* #ff9 shorthand → #FFFF99 */
}

.projectbox-second-background-7 {
    background-color: #87CEFA; /* LightSkyBlue */
}

.projectbox-second-background-8 {
    background-color: #FFDAB9; /* Peachpuff */
}

.projectbox-second-background-9 {
    background-color: #C0EAA6;
}

.projectbox-second-background-10 {
    background-color: #9AD4F6;
}

.projectbox-second-background-11 {
    background-color: #E6C6E6;
}

.projectbox-second-background-12 {
    background-color: #F4B8BF;
}

.projectbox-second-background-13 {
    background-color: #CED1FA;
}

.projectbox-second-background-14 {
    background-color: #E4E4E4;
}

.projectbox-second-background-15 {
    background-color: #D8BFD8; /* Thistle */
}

/* ===========================
   Dark Mode – projectbox-border-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-1 {
		border: 1px solid #B97A58;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-1 {
		border: 1px solid #B97A58;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-2 {
		border: 1px solid #1A6620;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-2 {
		border: 1px solid #1A6620;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-3 {
		border: 1px solid #3A628A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-3 {
		border: 1px solid #3A628A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-4 {
		border: 1px solid #A24A4A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-4 {
		border: 1px solid #A24A4A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-5 {
		border: 1px solid #7A52A8;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-5 {
		border: 1px solid #7A52A8;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-6 {
		border: 1px solid #CCAA33;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-6 {
		border: 1px solid #CCAA33;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-7 {
		border: 1px solid #3459A5;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-7 {
		border: 1px solid #3459A5;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-8 {
		border: 1px solid #C07052;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-8 {
		border: 1px solid #C07052;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-9 {
		border: 1px solid #5BA535;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-9 {
		border: 1px solid #5BA535;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-10 {
		border: 1px solid #3578A0;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-10 {
		border: 1px solid #3578A0;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-11 {
		border: 1px solid #9E6AA9;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-11 {
		border: 1px solid #9E6AA9;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-12 {
		border: 1px solid #A0525C;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-12 {
		border: 1px solid #A0525C;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-13 {
		border: 1px solid #6A6EC7;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-13 {
		border: 1px solid #6A6EC7;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-14 {
		border: 1px solid #777;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-14 {
		border: 1px solid #777;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-15 {
		border: 1px solid #B49BB4;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-15 {
		border: 1px solid #B49BB4;
	}
}

/* ===========================
   Dark Mode – projectbox-first-background-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-1 {
		background-color: #2A2620;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-1 {
		background-color: #2A2620;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-2 {
		background-color: #102A20;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-2 {
		background-color: #102A20;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-3 {
		background-color: #12202A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-3 {
		background-color: #12202A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-4 {
		background-color: #2A1A1C;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-4 {
		background-color: #2A1A1C;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-5 {
		background-color: #281A28;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-5 {
		background-color: #281A28;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-6 {
		background-color: #2A2818;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-6 {
		background-color: #2A2818;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-7 {
		background-color: #112028;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-7 {
		background-color: #112028;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-8 {
		background-color: #2A201A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-8 {
		background-color: #2A201A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-9 {
		background-color: #1B2715;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-9 {
		background-color: #1B2715;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-10 {
		background-color: #14222A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-10 {
		background-color: #14222A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-11 {
		background-color: #241A24;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-11 {
		background-color: #241A24;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-13 {
		background-color: #20202A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-13 {
		background-color: #20202A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-14 {
		background-color: #1A1A1A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-14 {
		background-color: #1A1A1A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-15 {
		background-color: #3A2D3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-15 {
		background-color: #3A2D3A;
	}
}

/* ===========================
   Dark Mode – projectbox-second-background-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-1 {
		background-color: #3A2D1A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-1 {
		background-color: #3A2D1A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-2 {
		background-color: #1E3A1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-2 {
		background-color: #1E3A1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-3 {
		background-color: #2A3444;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-3 {
		background-color: #2A3444;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-4 {
		background-color: #3A1E1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-4 {
		background-color: #3A1E1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-5 {
		background-color: #3A2D3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-5 {
		background-color: #3A2D3A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-6 {
		background-color: #4A4A19
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-6 {
		background-color: #4A4A19;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-7 {
		background-color: #1E2F44;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-7 {
		background-color: #1E2F44;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-8 {
		background-color: #3A2918;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-8 {
		background-color: #3A2918;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-9 {
		background-color: #293911;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-9 {
		background-color: #293911;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-10 {
		background-color: #1C2F3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-10 {
		background-color: #1C2F3A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-11 {
		background-color: #342434;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-11 {
		background-color: #342434;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-12 {
		background-color: #40252A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-12 {
		background-color: #40252A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-13 {
		background-color: #26283F;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-13 {
		background-color: #26283F;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-14 {
		background-color: #2A2A2A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-14 {
		background-color: #2A2A2A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-15 {
		background-color: #3A2D3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-15 {
		background-color: #3A2D3A;
	}
}</text>
      <sha1>7fx75ogm7hd0ttoty0sbbl4jbhoppoc</sha1>
    </revision>
    <revision>
      <id>2813361</id>
      <parentid>2813359</parentid>
      <timestamp>2026-06-07T00:28:17Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <minor />
      <comment>Protected &quot;[[Template:Projectbox/styles.css]]&quot;: Highly visible template ([Edit=Allow only curators and custodians] (indefinite) [Move=Allow only curators and custodians] (indefinite))</comment>
      <origin>2813359</origin>
      <model>sanitized-css</model>
      <format>text/css</format>
      <text bytes="16110" sha1="7fx75ogm7hd0ttoty0sbbl4jbhoppoc" xml:space="preserve">/* {{pp-template}} */
.projectbox {
	float: right;
	clear: right;
	margin: 5px 0 5px 5px;
	border-width: 1px;
	border-style: solid;
}
.projectbox-table {
	border: 0;
	border-collapse: collapse;
	border-spacing: 0;
	margin: 0;
	width: 228px;
}
.projectbox-image {
	border: 0;
	text-align: center;
	width: 45px;
	height: 45px;
}
.projectbox-text {
	border: 0;
	font-size: 8pt;
	line-height: 1.25em;
	padding: 0.5em;
}

/**
 * Mobile and small screens
 */
@media screen and ( max-width: 720px ) {
	.projectbox {
		margin: 5px 0;
	}
}

.projectbox-border-1 {
	border: 1px solid #E39C79;
}

.projectbox-border-2 {
	border: 1px solid #228B22; /* ForestGreen */
}

.projectbox-border-3 {
    border: 1px solid #4682B4; /* SteelBlue */
}

.projectbox-border-4 {
	border: 1px solid #CD5C5C; /* IndianRed */
}

.projectbox-border-5 {
	border: 1px solid #9370DB; /* MediumPurple */
}

.projectbox-border-6 {
    border: 1px solid #FFD700; /* Gold */
}

.projectbox-border-7 {
    border: 1px solid #4169E1; /* RoyalBlue */
}

.projectbox-border-8 {
    border: 1px solid #E78A69;
}

.projectbox-border-9 {
    border: 1px solid #71BE3F;
}

.projectbox-border-10 {
    border: 1px solid #4290BC;
}

.projectbox-border-11 {
    border: 1px solid #C289C3;
}

.projectbox-border-12 {
    border: 1px solid #C56B74;
}

.projectbox-border-13 {
    border: 1px solid #8488DC;
}

.projectbox-border-14 {
    border: 1px solid #AAAAAA;
}

.projectbox-border-15 {
    border: 1px solid #D8BFD8; /* Thistle */
}

.projectbox-first-background-1 {
    background-color: #FFFCF1;
}

.projectbox-first-background-2 {
    background-color: #F5FFFA; /* MintCream */
}

.projectbox-first-background-3 {
    background-color: #F0F8FF; /* AliceBlue */
}

.projectbox-first-background-4 {
    background-color: #FFE4E1; /* MistyRose */
}

.projectbox-first-background-5 {
    background-color: #FFF0F5; /* LavenderBlush */
}

.projectbox-first-background-6 {
	background-color: #FFFFF0; /* Ivory */
}

.projectbox-first-background-7 {
	background-color: #F0FFFF; /* Azure */
}

.projectbox-first-background-8 {
	background-color: #FFF5EE; /* Seashell */
}

.projectbox-first-background-9 {
	background-color: #F6FFF1;
}

.projectbox-first-background-10 {
	background-color: #F4FAFF;
}

.projectbox-first-background-11 {
	background-color: #FFF8FF;
}

.projectbox-first-background-12 {
    background-color: #FFF6F8;
}

.projectbox-first-background-13 {
	background-color: #F5F5FF;
}

.projectbox-first-background-14 {
	background-color: #FFFFFF;
}

.projectbox-first-background-15 {
	background-color: #D8BFD8; /* Thistle */
}

.projectbox-second-background-1 {
    background-color: #F5DEB3; /* Wheat */
}

.projectbox-second-background-2 {
    background-color: #90EE90; /* LightGreen */
}

.projectbox-second-background-3 {
    background-color: #B0C4DE; /* LightSteelBlue */
}

.projectbox-second-background-4 {
    background-color: #F08080; /* LightCoral */
}

.projectbox-second-background-5 {
    background-color: #D8BFD8; /* Thistle */
}

.projectbox-second-background-6 {
    background-color: #FFFF99; /* #ff9 shorthand → #FFFF99 */
}

.projectbox-second-background-7 {
    background-color: #87CEFA; /* LightSkyBlue */
}

.projectbox-second-background-8 {
    background-color: #FFDAB9; /* Peachpuff */
}

.projectbox-second-background-9 {
    background-color: #C0EAA6;
}

.projectbox-second-background-10 {
    background-color: #9AD4F6;
}

.projectbox-second-background-11 {
    background-color: #E6C6E6;
}

.projectbox-second-background-12 {
    background-color: #F4B8BF;
}

.projectbox-second-background-13 {
    background-color: #CED1FA;
}

.projectbox-second-background-14 {
    background-color: #E4E4E4;
}

.projectbox-second-background-15 {
    background-color: #D8BFD8; /* Thistle */
}

/* ===========================
   Dark Mode – projectbox-border-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-1 {
		border: 1px solid #B97A58;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-1 {
		border: 1px solid #B97A58;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-2 {
		border: 1px solid #1A6620;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-2 {
		border: 1px solid #1A6620;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-3 {
		border: 1px solid #3A628A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-3 {
		border: 1px solid #3A628A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-4 {
		border: 1px solid #A24A4A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-4 {
		border: 1px solid #A24A4A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-5 {
		border: 1px solid #7A52A8;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-5 {
		border: 1px solid #7A52A8;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-6 {
		border: 1px solid #CCAA33;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-6 {
		border: 1px solid #CCAA33;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-7 {
		border: 1px solid #3459A5;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-7 {
		border: 1px solid #3459A5;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-8 {
		border: 1px solid #C07052;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-8 {
		border: 1px solid #C07052;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-9 {
		border: 1px solid #5BA535;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-9 {
		border: 1px solid #5BA535;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-10 {
		border: 1px solid #3578A0;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-10 {
		border: 1px solid #3578A0;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-11 {
		border: 1px solid #9E6AA9;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-11 {
		border: 1px solid #9E6AA9;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-12 {
		border: 1px solid #A0525C;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-12 {
		border: 1px solid #A0525C;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-13 {
		border: 1px solid #6A6EC7;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-13 {
		border: 1px solid #6A6EC7;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-14 {
		border: 1px solid #777;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-14 {
		border: 1px solid #777;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-border-15 {
		border: 1px solid #B49BB4;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-border-15 {
		border: 1px solid #B49BB4;
	}
}

/* ===========================
   Dark Mode – projectbox-first-background-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-1 {
		background-color: #2A2620;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-1 {
		background-color: #2A2620;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-2 {
		background-color: #102A20;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-2 {
		background-color: #102A20;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-3 {
		background-color: #12202A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-3 {
		background-color: #12202A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-4 {
		background-color: #2A1A1C;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-4 {
		background-color: #2A1A1C;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-5 {
		background-color: #281A28;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-5 {
		background-color: #281A28;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-6 {
		background-color: #2A2818;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-6 {
		background-color: #2A2818;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-7 {
		background-color: #112028;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-7 {
		background-color: #112028;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-8 {
		background-color: #2A201A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-8 {
		background-color: #2A201A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-9 {
		background-color: #1B2715;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-9 {
		background-color: #1B2715;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-10 {
		background-color: #14222A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-10 {
		background-color: #14222A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-11 {
		background-color: #241A24;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-11 {
		background-color: #241A24;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-12 {
		background-color: #2A1C1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-13 {
		background-color: #20202A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-13 {
		background-color: #20202A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-14 {
		background-color: #1A1A1A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-14 {
		background-color: #1A1A1A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-first-background-15 {
		background-color: #3A2D3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-first-background-15 {
		background-color: #3A2D3A;
	}
}

/* ===========================
   Dark Mode – projectbox-second-background-*
   =========================== */

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-1 {
		background-color: #3A2D1A;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-1 {
		background-color: #3A2D1A;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-2 {
		background-color: #1E3A1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-2 {
		background-color: #1E3A1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-3 {
		background-color: #2A3444;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-3 {
		background-color: #2A3444;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-4 {
		background-color: #3A1E1E;
	}
}

@media screen and (prefers-color-scheme: dark) {
	html.skin-theme-clientpref-os .projectbox-second-background-4 {
		background-color: #3A1E1E;
	}
}

@media screen {
	html.skin-theme-clientpref-night .projectbox-second-background-5 {
		background-color: #3A2D3A;
	}
}

@media screen and (prefers-color-scheme: dark) {
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}</text>
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    <title>James Cameron and Color</title>
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        <username>Atcovi</username>
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      <text bytes="2448" sha1="0c4ojk96sj1f714fxkwxx1ximrw1nwo" xml:space="preserve">{{film}}
James Cameron's ''Avatar'' contains specific color palettes that aided in the creation of his world of Avatar. James Cameron created the world of Avatar and reached his audience by blending colors that appealed to the eye visually, as well as choosing colors that would elicit responses.
=== Color Palette For Avatar: Explanation===
''Avatar'' contains blues and greens primarily. Colors can have the power to evoke emotions in an audience. The yellow and oranges found in sunrises can evoke the feeing of happiness, energize an audience or even lift an audience's mood, just to give an example. &lt;ref&gt; 99designs.com &lt;/ref&gt;
===Other Colors &amp; Associations ===
* Purple, like the kind found on the mystical plants that the main characters in Avatar wander through and touch, is seen as a color that can &quot;represent magic, peace and power&quot;. &lt;ref&gt; www.verywellmind.com &lt;/ref&gt;
* Red, much like the red found on the Great Leonopteryx that Jake Sully used to claim the final victory, can represent a &quot;deep need or want inside people to take action&quot;. This sentiment rang true for Jake Sully, as his Leonopteryx helped the Navi clan decide to help him in his fight against the Sky People. &lt;ref&gt; www.colorpsychology.org &lt;/ref&gt;
* Green, similar to the pale green skin of the Navi in ''Avatar: The Way of Water'' , '''could''' represent &quot;peace, rest, and security&quot;. This idea would make sense due to the Navi enviornment, as well as the fact that they were a safe and untouched Navi clan that provided sanctuary to Jake Sully and his family. &lt;ref&gt; www.impactplus.com &lt;/ref&gt;
=== Family Significance &amp; Color and Mood ===
Avatar and Avatar The Way of Water emphasize family as a community. In the first film, Jake Sully is a loner that is accepted into the bright, new world of Pandora. The Indigo and California blues that appeared on screen created a calming sensation as he was accepted into an interconnected family. The Way of Water is similar due to the calm and wonderful feeling of enduring the impossible task of protecting a family.  &lt;ref&gt;[https://www.schemecolor.com/avatar-colors.]&lt;/ref&gt;
See links below for more information about James Cameron's work
[[Titanic (1997)]]
See below for more information about how colors in movies can affect mood.
[[Yellow in Film]]


== References ==
&lt;references /&gt;
* www.verywellmind.com
* www.colorpsychology.org
* www.impactplus.com
* https://www.schemecolor.com/avatar-colors.
[[Category:Cinema Aesthetics]]</text>
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[[Category:Wikiversitans]]</text>
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== Basic Scratch ==

@[[User:RailwayEnthusiast2025|RailwayEnthusiast2025]] Hi, i did some editing hope you like it Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:04, 31 October 2025 (UTC)

:@[[User:RailwayEnthusiast2025|RailwayEnthusiast2025]] Did you like it ? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 23:16, 1 November 2025 (UTC)
::I haven't viewed it yet, but I will check! —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 09:10, 2 November 2025 (UTC)
:::Good, but I already started making the steps for coding the project. Do you think you could expand in those parts? Thanks, —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 09:14, 2 November 2025 (UTC)

== Chess ==

Maybe this is something for you? [[Chess]] [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 22:00, 6 November 2025 (UTC)

:There is also a Wikiversity chess team &lt;span style=&quot;background-color: #aaffaa;&quot;&gt;created at [https://lichess.org/team/wikiversity Lichess.org].&lt;/span&gt; If you like, please join. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 10:22, 8 November 2025 (UTC)

== Changed strike through ==

Hi @[[User:RailwayEnthusiast2025|RailwayEnthusiast2025]] I made this edit : https://en.wikiversity.org/w/index.php?diff=2768157 assuming someone was fooling with your text. If you have done that by yourself, i appologize! Pls let me know? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 14:21, 10 November 2025 (UTC)

:Hi, I saw that you removed the strikethrough that Dan reverted and later re-instated it. What made you change your mind? Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:39, 18 November 2025 (UTC)

== Timezone project ==

Hi @[[User:RailwayEnthusiast2025|RailwayEnthusiast2025]] I translated a few lines for you, only a small section.of [[User:RailwayEnthusiast2025/Time zones]] if you like it annd want me to continue, just let me know, otherwise simply undo it in history. Grazie [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:19, 22 November 2025 (UTC)

:Thank you for the changes. Yes, I would appreciate some more translations on the project. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 17:21, 22 November 2025 (UTC)
::[[User:Harold Foppele/sandbox-5]]  fun :) [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 18:50, 22 November 2025 (UTC)
:::@[[User:CuriousWhistler|CuriousWhistler]] Finished translation. Pls look at the section Second hand. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:38, 23 November 2025 (UTC)
::::Was a nice excersition. 😄.  Are you going to join the WV Chess team? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 15:59, 23 November 2025 (UTC)</text>
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    <title>User:Renamed user b184b357650f0813a48b7fb823faee66/Wikiversity Newsletter</title>
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      <text bytes="654" sha1="6mbl3cdq7wl0y6kt4386t6nxzob7to9" xml:space="preserve">== The Wikiversity View (Issue 1) ==
__NOTOC__
=== Editor’s Welcome ===
Welcome to this inaugural edition of the Wikiversity View (also known as the Wikiversity Newsletter), created and written by [[User:CuriousWhistler|CuriousWhistler]] on a three-month basis and is open for suggestions from anyone. 

== Notices ==
For beginners: Please use the [[Wikiversity:Colloquium|Colloquium]] for questions about topics and help related to Wikiversity.

The [[Wikiversity:Help desk|help desk]] is closed as of the time of writing, so please ask questions about Wikiversity on the [[Wikiversity:Colloquium|Colloquium]] instead.

== References ==
&lt;references /&gt;</text>
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      <text bytes="498" sha1="i3i9u5gxdvin4zukupjzf02au3a8zoz" xml:space="preserve">'''Experimenting with micro:bit'' 
For this exercise, you will require a micro:bit. A micro:bit is a small codeable pocket-sized circuit board which you can code in Scratch. In this subpage, we will be experimenting with micro:bit and its features.
== Setting up your micro:bit ==
Setting up your micro:bit for Scratch will require you to download and install Scratch Link.&lt;ref&gt;{{Cite web|url=https://scratch.mit.edu/|title=Scratch - micro:bit|website=scratch.mit.edu|access-date=2025-10-27}}&lt;/ref&gt;</text>
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{{note|Likely other user accounts: [[User:HumbleBeauty]], [[User:Subtlevirtue]], [[B:User:MeekFavor]], [[User:AssumingNOTHING]], [[User:~2025-50652-2]], [[User:205.154.222.227]], [[User:2601:647:6512:E448:8D11:DA0B:3044:C676]], etc. See also [[Wikiversity:Request custodian action#Block of MarsSterlingTurner]].}}</text>
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    <title>User talk:Renamed user b184b357650f0813a48b7fb823faee66/Archive 1</title>
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      <text bytes="7451" sha1="qb88y87pp5mv1p0vze61mmw30s2s21x" xml:space="preserve">==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]],  RockTransport!'''|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Koavf|me personally]] if you would like some [[Help:Contents|help]].

Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.

We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.

To find your way around, check out:
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
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* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
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&lt;br clear=&quot;both&quot;/&gt;

To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! --—[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:32, 6 December 2024 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:32, 6 December 2024 (UTC)

:Thanks! [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:41, 9 December 2024 (UTC)

== Newsletter for wikiversity ==

Hi {{PAGENAME}},

I am a periodic visitor this wiki. I have been here on and off for years, but Still have not figured out how this place works. Recently I have tried to participate in the Colloquiem and I get the feeling, but I may be wrong, that there are only a handful of volunteers left here who are actively involved.

I don't want to put any pressure on you, I know we are all volunteers with real lives outside of wikimedia.  I am curious to find out what happened to your idea of publishing a newsletter?

Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 20:39, 15 January 2025 (UTC)

:It's still going on and I am contributing to it, the problem is I haven't really found some topics to cover. Also, if you wanted to contribute, feel free to, mainly because I don't really know what to cover. [[User:RockTransport|''Rock Transport'']] 😊 ([[User_talk:RockTransport|Talk page]])  20:53, 15 January 2025 (UTC)
::I hope you don't give up. I myself self have no idea what issuing on on this wiki, whois doing what, etc. I am interested in a wide-range of topics, but have not been able to find any on wikiversity which are currently active. There were some I tried, such as wiki debates, but the person who started it is no longer around. Intried to start a new weikidebate but I had a technical difficulty doing it.
::Sorry about this post, I have a misbehaving autocomplete and I must run now. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 21:04, 15 January 2025 (UTC)
:::Thanks for your reply [[User:RockTransport|''Rock Transport'']] 😊 ([[User_talk:RockTransport|Talk page]])  21:54, 15 January 2025 (UTC)
:::* correction to my previous post: Looks like the creator of the [[:Category:Wikidebates|wikidebate section]] is @[[User:Guy vandegrift]] who is still very active here, I think?
:::[[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 17 January 2025 (UTC)
::::Rock Transport: I hope Guy vandegrift does not mind being pinged by me. I don't like talking about other wikizens without letting them know that they are being discussed, something that has happened me several times. However, I have been criticized on several occasions by the ersonI have pinged. Whatdo you think? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 17 January 2025 (UTC)
:::::It depends for me, because there has to be context as to why they're being discussed. [[User:RockTransport|''Rock Transport'']] 😊 ([[User_talk:RockTransport|Talk page]])  17:32, 19 January 2025 (UTC)

== are you 2 Users? ==

Hi {{PAGENAME}}, are youth same user as [[User:RockTransport]] [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 20:35, 22 January 2025 (UTC)

:Yes I am, I just changed my username. [[User:RailwayEnthusiast2025|''RailwayEnthusiast2025'']] 😊 ([[User_talk:RailwayEnthusiast2025|Talk page]])  20:43, 22 January 2025 (UTC)

== &quot;I have interests in playing musical instruments,... the piano&quot; ==

Hi {{PAGENAME}},  we seem to share at least one interest -playing the piano. In my case it is as a student :-) [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 20:17, 26 January 2025 (UTC)

:Wow, I never knew you had an interest in playing the piano, what a coincidence! [[User:RailwayEnthusiast2025|RailwayEnthusiast2025]] ([[User talk:RailwayEnthusiast2025|Talk page]] - [[Special:Contributions|Contributions]]) 19:13, 27 January 2025 (UTC)
:By the way, what type of pieces do you play? I'm a beginner, so I'd love to know! [[User:RailwayEnthusiast2025|RailwayEnthusiast2025]] ([[User talk:RailwayEnthusiast2025|Talk page]] - [[Special:Contributions|Contributions]]) 19:33, 28 January 2025 (UTC)

== RailwayEnthusiast2025 ==

yo I'm goona help you bro [[User:Hatusnemiku|Hatusnemiku]] ([[User talk:Hatusnemiku|discuss]] • [[Special:Contributions/Hatusnemiku|contribs]]) 15:26, 29 September 2025 (UTC)

== Reviews ==

@[[User:RailwayEnthusiast2025|RailwayEnthusiast2025]] Yooooo did you have time to look at my pages? Made 4 sofar. Curious what you think of them :)[[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 21:33, 25 October 2025 (UTC)</text>
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    <title>PSYC322 - Adolescent Psychology</title>
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{{tertiary}}
{{psychology}}
{{course}}
'''PSYC322 - Psychology of Adolescence'''

== Assignments ==

=== Study Activities ===
'''Study activities?''' (10 out of 18 required) Recommended way:

* Write out the learning objectives and answer them.
* PRINT OUT NOTES BEFORE EXAMS. 
====More====
The study guides include a set of learning objectives pertaining to the material covered in the module.  These learning objectives are an item-by-item listing of the material that you need to know from the chapter. Seriously, objective #1 of chapter 1 corresponds with question #1 on the exam.  For this reason, I HIGHLY recommend that you write out answers to these.

;Step 1:  Answer the learning objectives

Use your textbook and the power point presentations to answer the learning objectives. The key to doing these is to write A LOT of information. Most test questions are not simple factual questions. Most test questions will require you to conceptually understand the material or apply the material to an entirely new situation. 

When answering the objectives, you want to pay attention to the verb used in the objective. 
*If it says to &quot;list&quot; just write a bullet point list that provides the requested information. 
*If it says to &quot;explain or describe&quot; you need to know the conceptual information behind that concept in the objective. You'll want to include LOTS of information in these learning objectives. I recommend summarizing the entire section of the book as it relates to that topic. Here’s an example:   ”List and describe the advantages of exercise.”  You might list, lowered blood pressure, decrease risk for obesity, and decreased likelihood of depression.   This is the list.  Next you need to describe the advantages.  For example, explain how exercise is related to a decreased likelihood of obesity.  Explain why exercise decreases the likelihood of depression.
*If it says to &quot;identify an example&quot; you need to list the concepts, define them, and then create your own examples. First, be sure to list all of the items that apply to that concept.  For example, if I said, “Identify an example of operant conditioning, “ you would list all four types of operant conditioning.  Then write a definition of each one, AND write your own example.  If you have difficulty creating an example, perhaps you don’t fully understand this concept.  Go back to the material, re-read it, and perhaps even reach out to me or the UTA/GTA.

;Step 2:  View the Review Session:

During the review session I will provide additional information.  I may tell you where to locate the information, what makes the concept difficult, provide examples, etc.  After viewing the session, be sure to go back to your study guide and add information or make changes.  Make sure that you’ve noted the things I told you to pay attention to.

;Step 3: Review your Learning Objectives:

After you have written out good information about each of the learning objectives, take some time to review what you’ve written.  Does the material make sense?  Are you understanding the concepts?  

You can do this when you have some down time.  While you are waiting for a class to start or waiting for an appointment, pull out your notes and review them.  It doesn’t have to take long.

;Step 4: Print the Study Guide:

I would print out your study guide prior to taking the exam so that you can reference the material you’ve written while you are taking your exam.  You will not be able to access these notes electronically while you are in SmarterProctoring.

I can’t stress enough the importance of completing the study guides.  I can’t imagine  how you would study without answering these.  How else would know what to study. 

=== Review Session ===
'''Review session?'''

* Notes must be content from the session, must cover ALL the content, must be handwritten. Do NOT write the answers, but write what she says. Ex:

Obj#2: Identify an example biological, cognitive, and socioemotional processes.

'''Correct Response''':  Be sure to write an example of all three of these concepts.  The example given on the exam will be one of the ones used in the text.

'''Incorrect Response''': Biological: physical changes in the body like height and weight gains, changes in the brain, and hormonal changes.  

=== Exercises ===
'''Exercises?''' (3 out of 4 required) Recommended way:

* Do them FIRST IN A GOOGLE DOC.

=== Exams ===
'''Exams?''' (4 out of 4 required) Recommended way:

* 50 multiple choice questions; 65 minutes; closes at 5pm EST, must start by 3pm EST. 

=== Extra Credit ===
'''Extra Credit?''' (Can do 10 of these for 40 points max.) Recommended way:

* Do them, they're discussion boards

== Subpages ==
;Content
'''Exam 1'''
* [[/Module 2: Introduction to Adolescent Psychology/]]
* [[/Module 3: Introduction to Puberty, Health and Biological Foundations/]]
* [[/Module 4: The Brain and Cognitive Development/]]
'''Exam 2'''
* [[/Module 5: Introduction: The Self, Identity, Emotion, and Personality/]]
* [[/Module 6: Gender/]]
* [[PSYC322 - Adolescent Psychology/Module 7: Adolescent Sexuality|Module 7: Adolescent Sexuality]]
* [[/Module 8: Moral Development/]]
'''Exam 3'''
* [[/Module 9: Families/]]
* [[/Module 10: Peers/]]
* [[PSYC322 - Adolescent Psychology/Module 11: Schools|Module 11: Schools]]
'''Exam 4'''

* [[PSYC322 - Adolescent Psychology/Module 12: Achievement, Work, and Careers|Module 12: Achievement, Work, and Careers]]
* [[PSYC322 - Adolescent Psychology/Module 13: Culture|Module 13: Culture]]
* [[PSYC322 - Adolescent Psychology/Module 14: Drugs, Delinquency, and Depression|Module 14: Drugs, Delinquency, and Depression]]
== Source ==
Modules 2 - 11 serve as student notes from the textbook, [https://www.mheducation.com/highered/product/adolescence-santrock.html?viewOption=student Santrock, J. W. (2025). ''Adolescence'' (19th ed.). McGraw Hill Higher Education]. Modules 12 - 14 serve as student notes from lectures by [https://www.odu.edu/directory/suzanne-morrow Professor Suzanne Morrow of Old Dominion University].
[[Category:PSYC322 - Adolescent Psychology]]</text>
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    <title>User:Renamed user b184b357650f0813a48b7fb823faee66/To-do</title>
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      <contributor>
        <username>XXBlackburnXx</username>
        <id>1202178</id>
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      <minor />
      <comment>XXBlackburnXx moved page [[User:CuriousWhistler/To-do]] to [[User:Renamed user b184b357650f0813a48b7fb823faee66/To-do]] without leaving a redirect: Automatically moved page while renaming the user &quot;[[Special:CentralAuth/CuriousWhistler|CuriousWhistler]]&quot; to &quot;[[Special:CentralAuth/Renamed user b184b357650f0813a48b7fb823faee66|Renamed user b184b357650f0813a48b7fb823faee66]]&quot;</comment>
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      <text bytes="448" sha1="bh8381hvt5jjk0ldko3w9gozagfn82y" xml:space="preserve">Below you can find a list of what I'm currently working on and what I am going to do/am doing on Wikiversity. Once I believe the task has been completed, I add &lt;nowiki&gt;{{Done}}&lt;/nowiki&gt; to it. If I haven't completed a project, then I will put &lt;nowiki&gt;{{Not done}}&lt;/nowiki&gt; to it.
#Finish translating pages in [[Scratch Coding Projects]] and perhaps move it to mainspace {{Not done}}
#Start writing a magazine/newsletter for Wikiversity {{Not done}}</text>
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    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
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      <comment>/* Conclusions */</comment>
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      <text bytes="66283" sha1="d9qtcmv61oqvltbytl7xt2wly6if5p2" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="66896" sha1="bptozuohm3emkqeiv4mjq2ku0z3dyeo" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>bptozuohm3emkqeiv4mjq2ku0z3dyeo</sha1>
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      <id>2813313</id>
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      <text bytes="77989" sha1="pl3we0i88a7pznb5lj8m5b5ebeqbbnd" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...

|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.618~&lt;sup&gt;-1&lt;/sup&gt;
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.437~&lt;sup&gt;-1&lt;/sup&gt;
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.382~&lt;sup&gt;-1&lt;/sup&gt;
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.357~&lt;sup&gt;-1&lt;/sup&gt;
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.325~&lt;sup&gt;-1&lt;/sup&gt;
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.288~&lt;sup&gt;-1&lt;/sup&gt;
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 3&lt;/math&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|0.270~&lt;sup&gt;-1&lt;/sup&gt;
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|0.247~&lt;sup&gt;-1&lt;/sup&gt;
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|0.236~&lt;sup&gt;-1&lt;/sup&gt;
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;2\pi/5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.230~&lt;sup&gt;-1&lt;/sup&gt;
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|0.221~&lt;sup&gt;-1&lt;/sup&gt;
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|0.208~&lt;sup&gt;−1&lt;/sup&gt;
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|0.201~&lt;sup&gt;−1&lt;/sup&gt;
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|0.191~&lt;sup&gt;−1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="81201" sha1="9b4bk9qfhhypmbu9liusir7tmx11nwz" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#0&lt;br&gt;&lt;br&gt;0&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|&lt;math&gt;\pi&lt;/math&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |#15&lt;br&gt;&lt;br&gt;30&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.135~&lt;sup&gt;-1&lt;/sup&gt;
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#1&lt;br&gt;&lt;br&gt;1&lt;sub&gt;0&lt;/sub&gt;
|𝞯
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#14&lt;br&gt;&lt;br&gt;29&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°{{Efn|In the 120-cell's isoclinic rotations the rotation arc-angle is 12° (1/30 of a circle), not the 15.5~° arc of the #1 edge chord. Regardless of which central planes are the invariant rotation planes, any 120-cell isoclinic rotation by 12° will take the great polygon in ''every'' central plane to a congruent great polygon in a Clifford parallel central plane that is 12° away. Adjacent Clifford parallel great polygons (of every kind) are completely disjoint, and their nearest vertices are connected by ''two'' 120-cell edges (#1 chords of arc-length 15.5~°). The 12° rotation angle is not the arc of any vertex-to-vertex chord in the 120-cell. It occurs only as the two equal angles between adjacent Clifford parallel central ''planes'',{{Efn|name=isoclinic}} and it is the separation between adjacent rotation planes in ''all'' the 120-cell's various isoclinic rotations (not only in its characteristic rotation).|name=12° rotation angle}}
|0.270~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|1&lt;sup&gt;-1&lt;/sup&gt;
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.136~&lt;sup&gt;-1&lt;/sup&gt;
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#2&lt;br&gt;&lt;br&gt;2&lt;sub&gt;0&lt;/sub&gt;
|{{Efn|name=#2 chord}}
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.618~&lt;sup&gt;-1&lt;/sup&gt;
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.437~&lt;sup&gt;-1&lt;/sup&gt;
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.382~&lt;sup&gt;-1&lt;/sup&gt;
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.357~&lt;sup&gt;-1&lt;/sup&gt;
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.325~&lt;sup&gt;-1&lt;/sup&gt;
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.288~&lt;sup&gt;-1&lt;/sup&gt;
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 3&lt;/math&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|0.270~&lt;sup&gt;-1&lt;/sup&gt;
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|0.247~&lt;sup&gt;-1&lt;/sup&gt;
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|0.236~&lt;sup&gt;-1&lt;/sup&gt;
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;2\pi/5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.230~&lt;sup&gt;-1&lt;/sup&gt;
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|0.221~&lt;sup&gt;-1&lt;/sup&gt;
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|0.208~&lt;sup&gt;−1&lt;/sup&gt;
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|0.201~&lt;sup&gt;−1&lt;/sup&gt;
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|0.191~&lt;sup&gt;−1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="81195" sha1="gvbxrx32fm5xu8jkonh2pzie9fd6419" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#0&lt;br&gt;&lt;br&gt;0&lt;sub&gt;0&lt;/sub&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|&lt;math&gt;\pi&lt;/math&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |#15&lt;br&gt;&lt;br&gt;30&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.135~&lt;sup&gt;-1&lt;/sup&gt;
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#1&lt;br&gt;&lt;br&gt;1&lt;sub&gt;0&lt;/sub&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#14&lt;br&gt;&lt;br&gt;29&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°{{Efn|In the 120-cell's isoclinic rotations the rotation arc-angle is 12° (1/30 of a circle), not the 15.5~° arc of the #1 edge chord. Regardless of which central planes are the invariant rotation planes, any 120-cell isoclinic rotation by 12° will take the great polygon in ''every'' central plane to a congruent great polygon in a Clifford parallel central plane that is 12° away. Adjacent Clifford parallel great polygons (of every kind) are completely disjoint, and their nearest vertices are connected by ''two'' 120-cell edges (#1 chords of arc-length 15.5~°). The 12° rotation angle is not the arc of any vertex-to-vertex chord in the 120-cell. It occurs only as the two equal angles between adjacent Clifford parallel central ''planes'',{{Efn|name=isoclinic}} and it is the separation between adjacent rotation planes in ''all'' the 120-cell's various isoclinic rotations (not only in its characteristic rotation).|name=12° rotation angle}}
|0.270~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.136~&lt;sup&gt;-1&lt;/sup&gt;
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#2&lt;br&gt;&lt;br&gt;2&lt;sub&gt;0&lt;/sub&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.437~&lt;sup&gt;-1&lt;/sup&gt;
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.382~&lt;sup&gt;-1&lt;/sup&gt;
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.357~&lt;sup&gt;-1&lt;/sup&gt;
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.325~&lt;sup&gt;-1&lt;/sup&gt;
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.288~&lt;sup&gt;-1&lt;/sup&gt;
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 3&lt;/math&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|0.270~&lt;sup&gt;-1&lt;/sup&gt;
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|0.247~&lt;sup&gt;-1&lt;/sup&gt;
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|0.236~&lt;sup&gt;-1&lt;/sup&gt;
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;2\pi/5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.230~&lt;sup&gt;-1&lt;/sup&gt;
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|0.221~&lt;sup&gt;-1&lt;/sup&gt;
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|0.208~&lt;sup&gt;−1&lt;/sup&gt;
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|0.201~&lt;sup&gt;−1&lt;/sup&gt;
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|0.191~&lt;sup&gt;−1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="80220" sha1="51rygltr5to58onxcgg6lpjlmfaw5vz" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#0&lt;br&gt;&lt;br&gt;0&lt;sub&gt;0&lt;/sub&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|&lt;math&gt;\pi&lt;/math&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |#15&lt;br&gt;&lt;br&gt;30&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.135~&lt;sup&gt;-1&lt;/sup&gt;
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#1&lt;br&gt;&lt;br&gt;1&lt;sub&gt;0&lt;/sub&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#14&lt;br&gt;&lt;br&gt;29&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|0.270~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.136~&lt;sup&gt;-1&lt;/sup&gt;
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#2&lt;br&gt;&lt;br&gt;2&lt;sub&gt;0&lt;/sub&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.357~&lt;sup&gt;-1&lt;/sup&gt;
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.325~&lt;sup&gt;-1&lt;/sup&gt;
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.288~&lt;sup&gt;-1&lt;/sup&gt;
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 3&lt;/math&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|0.270~&lt;sup&gt;-1&lt;/sup&gt;
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|0.247~&lt;sup&gt;-1&lt;/sup&gt;
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|0.236~&lt;sup&gt;-1&lt;/sup&gt;
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;2\pi/5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.230~&lt;sup&gt;-1&lt;/sup&gt;
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|0.221~&lt;sup&gt;-1&lt;/sup&gt;
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|0.208~&lt;sup&gt;−1&lt;/sup&gt;
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|0.201~&lt;sup&gt;−1&lt;/sup&gt;
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|0.191~&lt;sup&gt;−1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="80220" sha1="3j63z69givj4x78dwnvfe2mqpu19e7y" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#0&lt;br&gt;&lt;br&gt;0&lt;sub&gt;0&lt;/sub&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|&lt;math&gt;\pi&lt;/math&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |#15&lt;br&gt;&lt;br&gt;30&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.135~&lt;sup&gt;-1&lt;/sup&gt;
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#1&lt;br&gt;&lt;br&gt;1&lt;sub&gt;0&lt;/sub&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#14&lt;br&gt;&lt;br&gt;29&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|0.270~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.136~&lt;sup&gt;-1&lt;/sup&gt;
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#2&lt;br&gt;&lt;br&gt;2&lt;sub&gt;0&lt;/sub&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 3&lt;/math&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|0.270~&lt;sup&gt;-1&lt;/sup&gt;
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|0.247~&lt;sup&gt;-1&lt;/sup&gt;
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|0.236~&lt;sup&gt;-1&lt;/sup&gt;
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;2\pi/5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.230~&lt;sup&gt;-1&lt;/sup&gt;
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|0.221~&lt;sup&gt;-1&lt;/sup&gt;
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|0.208~&lt;sup&gt;−1&lt;/sup&gt;
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|0.201~&lt;sup&gt;−1&lt;/sup&gt;
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|0.191~&lt;sup&gt;−1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="80178" sha1="rtx2dwh5fz8riize21r9p1dp6wugbrg" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#0&lt;br&gt;&lt;br&gt;0&lt;sub&gt;0&lt;/sub&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|&lt;math&gt;\pi&lt;/math&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |#15&lt;br&gt;&lt;br&gt;30&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.135~&lt;sup&gt;-1&lt;/sup&gt;
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#1&lt;br&gt;&lt;br&gt;1&lt;sub&gt;0&lt;/sub&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#14&lt;br&gt;&lt;br&gt;29&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|0.270~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.136~&lt;sup&gt;-1&lt;/sup&gt;
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#2&lt;br&gt;&lt;br&gt;2&lt;sub&gt;0&lt;/sub&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|0.201~&lt;sup&gt;−1&lt;/sup&gt;
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|0.191~&lt;sup&gt;−1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="80156" sha1="duz8l01f4fcezok2ka00mfn0t1d9blg" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#0&lt;br&gt;&lt;br&gt;0&lt;sub&gt;0&lt;/sub&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|&lt;math&gt;\pi&lt;/math&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |#15&lt;br&gt;&lt;br&gt;30&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.135~&lt;sup&gt;-1&lt;/sup&gt;
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#1&lt;br&gt;&lt;br&gt;1&lt;sub&gt;0&lt;/sub&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#14&lt;br&gt;&lt;br&gt;29&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|0.270~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.136~&lt;sup&gt;-1&lt;/sup&gt;
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#2&lt;br&gt;&lt;br&gt;2&lt;sub&gt;0&lt;/sub&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="80154" sha1="5j7guwvmn4im4kgn7xjeapbylt3l936" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#0&lt;br&gt;&lt;br&gt;0&lt;sub&gt;0&lt;/sub&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|30&lt;sub&gt;0&lt;/sub&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |#15&lt;br&gt;&lt;br&gt;30&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.135~&lt;sup&gt;-1&lt;/sup&gt;
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#1&lt;br&gt;&lt;br&gt;1&lt;sub&gt;0&lt;/sub&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#14&lt;br&gt;&lt;br&gt;29&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|0.270~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.136~&lt;sup&gt;-1&lt;/sup&gt;
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#2&lt;br&gt;&lt;br&gt;2&lt;sub&gt;0&lt;/sub&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>5j7guwvmn4im4kgn7xjeapbylt3l936</sha1>
    </revision>
    <revision>
      <id>2813321</id>
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      <timestamp>2026-06-06T19:58:56Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Complementary chord pairs */</comment>
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      <text bytes="80137" sha1="c7xljezard6kqfohz3u6w5230bh895a" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#0&lt;br&gt;&lt;br&gt;0&lt;sub&gt;0&lt;/sub&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|30&lt;sub&gt;0&lt;/sub&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |#15&lt;br&gt;&lt;br&gt;30&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#1&lt;br&gt;&lt;br&gt;1&lt;sub&gt;0&lt;/sub&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|29&lt;sub&gt;0&lt;/sub&gt;
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#14&lt;br&gt;&lt;br&gt;29&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|0.270~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.270~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#2&lt;br&gt;&lt;br&gt;2&lt;sub&gt;0&lt;/sub&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>c7xljezard6kqfohz3u6w5230bh895a</sha1>
    </revision>
    <revision>
      <id>2813322</id>
      <parentid>2813321</parentid>
      <timestamp>2026-06-06T20:01:33Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Complementary chord pairs */</comment>
      <origin>2813322</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="80137" sha1="4dmlqnvu06y0elc470mr384r9oxgkgq" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#0&lt;br&gt;&lt;br&gt;0&lt;sub&gt;0&lt;/sub&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|30&lt;sub&gt;0&lt;/sub&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |#15&lt;br&gt;&lt;br&gt;30&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#1&lt;br&gt;&lt;br&gt;1&lt;sub&gt;0&lt;/sub&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|29&lt;sub&gt;0&lt;/sub&gt;
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#14&lt;br&gt;&lt;br&gt;29&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.270~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#2&lt;br&gt;&lt;br&gt;2&lt;sub&gt;0&lt;/sub&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Complementary chord pairs */</comment>
      <origin>2813323</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="80137" sha1="etytvkywvytix59nmyspfdid08lfssw" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#0&lt;br&gt;&lt;br&gt;0&lt;sub&gt;0&lt;/sub&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|30&lt;sub&gt;0&lt;/sub&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |#15&lt;br&gt;&lt;br&gt;30&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#1&lt;br&gt;&lt;br&gt;1&lt;sub&gt;0&lt;/sub&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|29&lt;sub&gt;0&lt;/sub&gt;
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#14&lt;br&gt;&lt;br&gt;29&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#2&lt;br&gt;&lt;br&gt;2&lt;sub&gt;0&lt;/sub&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |#13&lt;br&gt;&lt;br&gt;28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#3&lt;br&gt;&lt;br&gt;3&lt;sub&gt;0&lt;/sub&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;br&gt;&lt;br&gt;27&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |#3&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;4&lt;sub&gt;0&lt;/sub&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#12&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;26&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#4&lt;br&gt;&lt;br&gt;5&lt;sub&gt;0&lt;/sub&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;br&gt;&lt;br&gt;25&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#4&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;6&lt;sub&gt;0&lt;/sub&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#11&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;24&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;7&lt;sub&gt;0&lt;/sub&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;23&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |#5&lt;br&gt;&lt;br&gt;8&lt;sub&gt;0&lt;/sub&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;br&gt;&lt;br&gt;22&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#5&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;9&lt;sub&gt;0&lt;/sub&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#10&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;21&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;10&lt;sub&gt;0&lt;/sub&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;20&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |#6&lt;br&gt;&lt;br&gt;11&lt;sub&gt;0&lt;/sub&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#9&lt;br&gt;&lt;br&gt;19&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;12&lt;sub&gt;0&lt;/sub&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#8&lt;br&gt;&lt;br&gt;18&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;13&lt;sub&gt;0&lt;/sub&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |#8&lt;sup&gt;−&lt;/sup&gt;&lt;br&gt;&lt;br&gt;17&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |#6&lt;sup&gt;++&lt;/sup&gt;&lt;br&gt;&lt;br&gt;14&lt;sub&gt;0&lt;/sub&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;sup&gt;+&lt;/sup&gt;&lt;br&gt;&lt;br&gt;16&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |#7&lt;br&gt;&lt;br&gt;15&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <id>2813324</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="79707" sha1="m794o8hssud6ldsxbf5yy8hz8l3vmtx" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|30&lt;sub&gt;0&lt;/sub&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_30&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|29&lt;sub&gt;0&lt;/sub&gt;
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_29&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_28&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_27&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_26&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_25&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_24&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_23&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_22&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_21&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_10&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_20&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_11&lt;/math&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_19&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_12&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_18&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_13&lt;/math&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_17&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_14&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_16&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_15&lt;/math&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_15&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>m794o8hssud6ldsxbf5yy8hz8l3vmtx</sha1>
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      <contributor>
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      <text bytes="79751" sha1="kdo16l24s2utwvcltmfrgdjohyeh20t" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|30&lt;sub&gt;0&lt;/sub&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|29&lt;sub&gt;0&lt;/sub&gt;
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.138~&lt;sup&gt;-1&lt;/sup&gt;
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>kdo16l24s2utwvcltmfrgdjohyeh20t</sha1>
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    <revision>
      <id>2813332</id>
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      <timestamp>2026-06-06T22:12:17Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="79752" sha1="1pfpc1qsbc9iewpx1h6sdco7sxxugk1" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|30&lt;sub&gt;0&lt;/sub&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|29&lt;sub&gt;0&lt;/sub&gt;
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|28&lt;sub&gt;0&lt;/sub&gt;
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>1pfpc1qsbc9iewpx1h6sdco7sxxugk1</sha1>
    </revision>
    <revision>
      <id>2813333</id>
      <parentid>2813332</parentid>
      <timestamp>2026-06-06T22:13:37Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="79752" sha1="3f2kkxittgs8iou4azh17ai2hgqsdlu" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|30&lt;sub&gt;0&lt;/sub&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|29&lt;sub&gt;0&lt;/sub&gt;
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|28&lt;sub&gt;0&lt;/sub&gt;
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|3&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°{{Efn|name=dihedral}}
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.142~&lt;sup&gt;-1&lt;/sup&gt;
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="79725" sha1="noj0q9nmemnzlstplnsy2ybao1ylkgd" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|30&lt;sub&gt;0&lt;/sub&gt;
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|1&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|29&lt;sub&gt;0&lt;/sub&gt;
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|2&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|28&lt;sub&gt;0&lt;/sub&gt;
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|4&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.144~&lt;sup&gt;-1&lt;/sup&gt;
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|5&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|6&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="79592" sha1="amg4rdvrebveefyhdw4swv2idqly7yv" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.146~&lt;sup&gt;-1&lt;/sup&gt;
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Complementary chord pairs */</comment>
      <origin>2813339</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="79579" sha1="54536ka8v8ayw6iwlstkbtl7mmpvusa" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|7&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|9&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|11&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|13&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|15&lt;sub&gt;0&lt;/sub&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <contributor>
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      <text bytes="79511" sha1="rooe78kshi9sxxf64sqqmretebri4gz" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.148~&lt;sup&gt;-1&lt;/sup&gt;
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.153~&lt;sup&gt;-1&lt;/sup&gt;
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.156~&lt;sup&gt;-1&lt;/sup&gt;
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.161~&lt;sup&gt;-1&lt;/sup&gt;
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>rooe78kshi9sxxf64sqqmretebri4gz</sha1>
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      <id>2813341</id>
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      <text bytes="79459" sha1="k9rquz70e6p4sm1tdxqu568u9f4ce3k" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|8&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.091~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>k9rquz70e6p4sm1tdxqu568u9f4ce3k</sha1>
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      <id>2813342</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="79481" sha1="2zjzy9xplj0cfyl4n7cjjpyvccb2020" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.091~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>2zjzy9xplj0cfyl4n7cjjpyvccb2020</sha1>
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      <text bytes="79481" sha1="451prnlbzi1crhhyao72srgt4qxcep3" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.091~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>451prnlbzi1crhhyao72srgt4qxcep3</sha1>
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    <revision>
      <id>2813344</id>
      <parentid>2813343</parentid>
      <timestamp>2026-06-06T22:38:51Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Complementary chord pairs */</comment>
      <origin>2813344</origin>
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      <format>text/x-wiki</format>
      <text bytes="79481" sha1="8323pauz7ia7c91eyyq08ycs7ve0mm6" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.676~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.165~&lt;sup&gt;-1&lt;/sup&gt;
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>8323pauz7ia7c91eyyq08ycs7ve0mm6</sha1>
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      <contributor>
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      <text bytes="79468" sha1="na410j3tb0101bqul6w14czd9wftguq" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.676~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.640~
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>na410j3tb0101bqul6w14czd9wftguq</sha1>
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    <revision>
      <id>2813347</id>
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      <timestamp>2026-06-06T22:41:43Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="79488" sha1="fzaspnnv5ld6n9xtuk2agn9q3iw94ib" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.676~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.640~
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>fzaspnnv5ld6n9xtuk2agn9q3iw94ib</sha1>
    </revision>
    <revision>
      <id>2813348</id>
      <parentid>2813347</parentid>
      <timestamp>2026-06-06T22:42:27Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Complementary chord pairs */</comment>
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      <text bytes="79489" sha1="a6ah42t181jif0qmpksd0qw4s0bjgq2" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.676~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.640~
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|&lt;math&gt;2\pi / 5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.167~&lt;sup&gt;-1&lt;/sup&gt;
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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    <revision>
      <id>2813349</id>
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      <text bytes="79476" sha1="0aignmwe5v5jqpzgtvvvtd3exz7j75x" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.676~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.640~
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|&lt;math&gt;2\pi / 5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.618~
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|12&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.171~&lt;sup&gt;-1&lt;/sup&gt;
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="79449" sha1="i4xm776uo1lgui9nqq1kuwqinqkxoak" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.676~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.640~
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|&lt;math&gt;2\pi / 5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.618~
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.581~
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.178~&lt;sup&gt;-1&lt;/sup&gt;
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="79436" sha1="10mkqf4720mjj2kqmdwxw9o4aq58v69" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
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|-
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|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
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|-
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|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
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|-
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|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.676~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.640~
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|&lt;math&gt;2\pi / 5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.618~
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.581~
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.520~
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.182~&lt;sup&gt;-1&lt;/sup&gt;
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="79423" sha1="lxlzt0bc98eypqrlrzospck83kgk12a" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.676~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.640~
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|&lt;math&gt;2\pi / 5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.618~
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.581~
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.520~
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.480~
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|0.191~&lt;sup&gt;-1&lt;/sup&gt;
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>lxlzt0bc98eypqrlrzospck83kgk12a</sha1>
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    <revision>
      <id>2813353</id>
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      <timestamp>2026-06-06T22:48:15Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="79410" sha1="3zaszopvzi8pkbuh0qb1o1ds5h4840t" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|&lt;math&gt;\pi / 5&lt;/math&gt;
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|&lt;math&gt;4\pi / 5&lt;/math&gt;
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|&lt;small&gt;&lt;math&gt;\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|&lt;small&gt;&lt;math&gt;2\pi / 3&lt;/math&gt;&lt;/small&gt;
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.676~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|10&lt;sub&gt;0&lt;/sub&gt;
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.640~
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|&lt;math&gt;2\pi / 5&lt;/math&gt;
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|&lt;math&gt;3\pi / 5&lt;/math&gt;
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.618~
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.581~
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.520~
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|14&lt;sub&gt;0&lt;/sub&gt;
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.480~
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|&lt;math&gt;\pi / 2&lt;/math&gt;
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="79661" sha1="iv0hh6cg3tt42k3ejdo7l3slv2j9g4d" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest, central section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a non-uniform 60-point [[w:Rhombicosidodecahedron|rhombicosidodecahedron.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of great circle polygons and vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ&lt;sup&gt;−2&lt;/sup&gt;√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1&lt;sub&gt;0&lt;/sub&gt; − 16&lt;sub&gt;0&lt;/sub&gt;|ps=, but 14&lt;sub&gt;0&lt;/sub&gt; and 16&lt;sub&gt;0&lt;/sub&gt; are congruent opposing sections and 15&lt;sub&gt;0&lt;/sub&gt; opposes itself; there are 29 non-point sections, denoted 1&lt;sub&gt;0&lt;/sub&gt; − 29&lt;sub&gt;0&lt;/sub&gt;, in 15 opposing pairs.}}
|-
! colspan=&quot;4&quot; |Short chord
! colspan=&quot;2&quot; |Great circle polygons
!Rotation
! colspan=&quot;4&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|{{radic|0}}
|{{radic|0}}
|{{radic|0}}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|{{radic|4}}
|{{radic|4}}
|{{radic|4}}
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|0°
|0
|0
|180°
|2
|2
|- style=&quot;background: palegreen;&quot; |
|0
|0
|&lt;small&gt;&lt;math&gt;0\times\zeta&lt;/math&gt;&lt;/small&gt;
|2
|7.405~
|&lt;small&gt;&lt;math&gt;2\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|{{radic|0.𝜀}}{{Efn|name=fractional square roots}}
|{{radic|0.𝜀}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^4}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}}
|{{radic|3.93~}}
|{{radic|3.93~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|15.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;1 / \phi^2\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|164.5~°
|1.982~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1
|&lt;small&gt;&lt;math&gt;1\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.982~
|7.337~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|{{radic|0.19~}}
|{{radic|0.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|{{radic|3.81~}}
|{{radic|3.81~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|25.2~°
|0.437~
|&lt;small&gt;&lt;math&gt;1 / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|154.8~°
|1.952~
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.618~
|&lt;small&gt;&lt;math&gt;\phi\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.952~
|7.226~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt; {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=&lt;br&gt;‡ For simplicity we omit the value of &lt;math&gt;a&lt;/math&gt; whenever it is not mononomial in &lt;math&gt;\chi&lt;/math&gt;, &lt;math&gt;\psi&lt;/math&gt; and &lt;math&gt;\phi&lt;/math&gt;.}}
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|{{radic|0.𝚫}}
|{{radic|0.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|{{radic|3.𝚽}}
|{{radic|3.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|36°
|0.618~
|&lt;small&gt;&lt;math&gt;1 / \phi&lt;/math&gt;&lt;/small&gt;
|144°
|1.902~
|&lt;small&gt;&lt;math&gt;1+1/{\phi^2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|0.618~
|2.288~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.902~
|7.0425
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|{{radic|0.5}}
|{{radic|0.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{1/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|{{radic|3.5}}
|{{radic|3.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{7/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|41.4~°
|0.707~
|&lt;small&gt;&lt;math&gt;\sqrt{2}/2&lt;/math&gt;&lt;/small&gt;
|138.6~°
|1.871~
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|2.618~
|&lt;small&gt;&lt;math&gt;\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.871~
|6.927~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{7}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|{{radic|0.57~}}
|{{radic|0.57~}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/{2\phi^2}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|{{radic|3.43~}}
|{{radic|3.43~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^4/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|44.5~°
|0.757~
|&lt;small&gt;&lt;math&gt;\sqrt{3} / \phi\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|135.5~°
|1.851~
|&lt;small&gt;&lt;math&gt;\phi^2 / \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen;&quot; |
|0.757~
|2.803~
|&lt;small&gt;&lt;math&gt;\phi\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.851~
|6.854~
|&lt;small&gt;&lt;math&gt;\phi^4\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|{{radic|0.69~}}
|{{radic|0.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|{{radic|3.31~}}
|{{radic|3.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \sqrt{5}/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|49.1~°
|0.831~
|
|130.9~°
|1.819~
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|3.078~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.819~
|6.735~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|{{radic|0.88~}}
|{{radic|0.88~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|{{radic|3.12~}}
|{{radic|3.12~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \psi/{2\phi}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|56°
|0.939~
|
|124°
|1.766~
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|3.477~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.766~
|6.538~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:&lt;br&gt;
{{indent|4}}&lt;math&gt;11/\chi = \psi&lt;/math&gt;
&lt;br&gt;
{{indent|4}}&lt;math&gt;\chi=(3\sqrt{5}+1)/2 \approx 3.854~&lt;/math&gt;
{{indent|4}}&lt;math&gt;\psi=(3\sqrt{5}-1)/2 \approx 2.854~&lt;/math&gt;}}
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|{{radic|1}}
|{{radic|1}}
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|{{radic|3}}
|{{radic|3}}
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|60°
|1
|
|120°
|1.732~
|
|- style=&quot;background: palegreen;&quot; |
|1
|3.702~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.732~
|6.413~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{6}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|{{radic|1.19~}}
|{{radic|1.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|{{radic|2.81~}}
|{{radic|2.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \chi/2\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|66.1~°
|1.091~
|
|113.9~°
|1.676~
|
|- style=&quot;background: gainsboro;&quot; |
|1.676~
|4.041~
|&lt;small&gt;&lt;math&gt;\sqrt{\chi/\phi^3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.676~
|6.205~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|{{radic|1.31~}}
|{{radic|1.31~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|{{radic|2.69~}}
|{{radic|2.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{4 - \phi^2/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|69.8~°
|1.144~
|&lt;small&gt;&lt;math&gt;\phi/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|110.2~°
|1.640~
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|4.236~
|&lt;small&gt;&lt;math&gt;\phi^3\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.640~
|6.074~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|{{radic|1.𝚫}}
|{{radic|1.𝚫}}
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|{{radic|2.𝚽}}
|{{radic|2.𝚽}}
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|72°
|1.176~
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5}/\phi}&lt;/math&gt;&lt;/small&gt;
|108°
|1.618~
|&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot; |
|1.176~
|4.353~
|&lt;small&gt;&lt;math&gt;\sqrt{2\phi^3\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.618~
|5.991~
|&lt;small&gt;&lt;math&gt;\phi^3\sqrt{2}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|{{radic|1.5}}
|{{radic|1.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{3/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅{{Efn|name=isocline circumference}}&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|{{radic|2.5}}
|{{radic|2.5}}
|&lt;small&gt;&lt;math&gt;\sqrt{5/2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|75.5~°
|1.224~
|
|104.5~°
|1.581~
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|4.535~
|&lt;small&gt;&lt;math&gt;\phi^2\sqrt{3}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.581~
|5.854~
|&lt;small&gt;&lt;math&gt;\sqrt{5\phi^4}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|{{radic|1.69~}}
|{{radic|1.69~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}(9-\sqrt{5})}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|{{radic|2.31~}}
|{{radic|2.31~}}
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|81.1~°
|1.300~
|&lt;small&gt;&lt;math&gt;\tfrac{1}{2}\sqrt{9-\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|98.9~°
|1.520~
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|4.815~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.520~
|5.626~
|&lt;small&gt;&lt;math&gt;\sqrt{\psi\phi^5}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|{{radic|0.81~}}
|{{radic|0.81~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2\phi\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|{{radic|2.19~}}
|{{radic|2.19~}}
|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{11-\sqrt{5}}{4}}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|84.5~°
|1.345~
|
|95.5~°
|1.480~
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|4.980~
|&lt;small&gt;&lt;math&gt;\sqrt{\phi^5\sqrt{5}}\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.480~
|5.480~
|&lt;small&gt;&lt;math&gt;\text{‡}\times\zeta&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|{{radic|2}}
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|{{radic|2}}
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|90°
|1.414~
|
|90°
|1.414~
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|1.414~
|5.236~
|&lt;small&gt;&lt;math&gt;2\phi^2\times\zeta&lt;/math&gt;&lt;/small&gt;
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 12-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to Clifford polygon sequences in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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    <title>Intuitive Calculus</title>
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      <text bytes="14832" sha1="c18tj140xcd15xdpmbe7vwn0340ibwg" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
{{juststarted}}
{{contrib-creator|[[User:Atcovi|Atcovi]]}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maximum is). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 [Fermet's Theorem] ===

* Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read [https://www.osti.gov/servlets/purl/400027 this] for more about the FBI's decision to digitalize fingerprint files and the process behind it.
* ''[expand upon Fermat's optimization? Use the PDF?]''
* '''Fermet's Theorem =''' If a real-valued function, &lt;math&gt;f(x)&lt;/math&gt;, is differentiable&lt;ref&gt;function has a well-defined, smooth slope at every single point&lt;/ref&gt; in an interval &lt;math&gt;(a, b)&lt;/math&gt; and &lt;math&gt;f(x)&lt;/math&gt; has a maximum OR minimum at &lt;math&gt;c&lt;/math&gt; ∈ &lt;math&gt;(a, b)&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;f'(c)&lt;/math&gt; = &lt;math display=&quot;inline&quot;&gt;0&lt;/math&gt;&lt;ref&gt;{{Cite web|url=https://old.maa.org/press/periodicals/convergence/fermat-s-method-for-finding-maxima-and-minima-a-mini-primary-source-project-for-calculus-1-students|title=Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students {{!}} Mathematical Association of America|website=old.maa.org|access-date=2026-05-21}}&lt;/ref&gt;.
** Explanation of ∈: essentially &quot;belongs to/inside/a member of.&quot; For example, &lt;math&gt;c&lt;/math&gt; ∈ &lt;math&gt;(a, b)&lt;/math&gt; → &quot;the number c&lt;math&gt;&lt;/math&gt; is inside the interval between &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt;&quot;.

=== 5/23/2026 [Logarithms] ===
''[insert logarithms introduction/lesson]''

log(''a'' x ''b'') = log ''a'' + log ''b''

Multiply two numbers together, take the log = answer is the SUM of their individual logs. Logarithms are like an &quot;undo&quot; tool. They &quot;undo&quot; the mathematical operations done by exponential functions, and the relationship between logarithms and exponential functions is reciprocal. 

* ''e''  = 2.71828... similar to π in circles&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=136}}&lt;/ref&gt;. See [https://simple.wikipedia.org/wiki/E_(mathematical_constant) e (mathematical constant)] (simple-wiki) &amp; [[w:Natural logarithm]] (wikipedia). The rate of change of ''e''&lt;sup&gt;x&lt;/sup&gt; is ''e''&lt;sup&gt;x&lt;/sup&gt;. The rate of exponential growth is proportional to the function's current level&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=137}}&lt;/ref&gt;. An example to illustrate this is the following: as a microphone picks up a noise that increases in volume (perhaps the source of the sound is moving closer to the microphone), the loudspeaker amplifies the noise at a constant, exponential rate ''in proportional'' (NOT equal) to the noise it is picking up through the microphone.

=== 5/27/2026 [Derivatives] ===
[[File:2020-03-25 00 08 15 A Five Cheese Pizza Hot Pocket after being heated in the Franklin Farm section of Oak Hill, Fairfax County, Virginia.jpg|thumb|When looking at how many ''more'' calories I will consume per infinitesimally small bite of the hot pocket, we are assessing the derivative of the hot pocket's calories. 

Yes, this may not be practical, but hopefully bringing food into the 'equation' will help you understand the concept of derivatives better.]]

* What is the definition of a '''derivative'''? Essentially the rate of change: ''dy/dx''. An example of a derivative is [[PlanetPhysics/Acceleration|acceleration]]. Another example of a derivative is the following question: how many calories will I consume per bite of a hot pocket (each bite being infinitesimally small)?

The question posed by the book is as follows: ''how do we define the slope when the slope keeps changing?''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=143}}&lt;/ref&gt;

Shifting our mindset from [[Speak Math Now!|algebra]]: In calculus, the rate of change is ''not'' constant, as the IV changes (and is therefore regarded as a '''function'''). We go from Δy/Δx [set rate of change] → ''dy/dx'' [infinitesimally tiny, varied changes]. 

So instead of thinking of the hourly rate for a cashier as a set number (let's say $16/hr), we should think of the $16/hr as a ''constant'' function. This is going to pay off in calculus as we deal with rates of changes that are not always 'set in stone', or constant. For example, measuring a horse's total speed in a [[w:Horse_racing|horse race]] is not going to be a constant, set number - it will be a function with a constantly changing rate. For this specific example:

* '''x''' = time
* '''y''' = speed
* '''dy/dx''' = rate of change of horse's speed with respect to time (think of it as: &quot;rate of change of [y] in respect to [x]&quot;).

=== 6/6/2026 ===
''background info...''

A '''definite''' '''integral''', in calculus, is the generalized area under the curved function&lt;ref&gt;{{Cite web|url=https://openstax.org/books/calculus-volume-1/pages/5-2-the-definite-integral|title=5.2 The Definite Integral - Calculus Volume 1 {{!}} OpenStax|last=Strang|first=Gilbert|last2=Herman|first2=Edwin “Jed”|date=2016-03-30|website=openstax.org|language=English|access-date=2026-06-06}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/definite-integral-as-the-limit-of-a-riemann-sum|title=Khan Academy|website=www.khanacademy.org|language=en|access-date=2026-06-06}}&lt;/ref&gt;.

* '''Area function''' (calculus) - Accumulated area under a curved line on an ''xy'' graph from point ''a'' to point ''x'' [upper bound that can be moved as opposed to point ''b'']. [https://www.youtube.com/watch?v=6WfetaviTrQ YT video].

TO-DO [6/6/2026]: provide example problem of area function modeling of YT video, do the work, take a pic and upload.

== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]
'''[[w:Pierre_de_Fermat|Pierre de Fermat]]'''

* [https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf ''Fermat’s Method for Finding Maxima and Minima'']- Kenneth M Monks (2023)
'''Other'''

* [[w:Glossary_of_mathematical_symbols|Glossary of mathematical symbols]]

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
{{reflist}}
[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
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      <text bytes="14869" sha1="fjjev7gve0nqs6m8s6p9mm40i8u9tcj" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
{{juststarted}}
{{contrib-creator|[[User:Atcovi|Atcovi]]}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maximum is). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 [Fermet's Theorem] ===

* Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read [https://www.osti.gov/servlets/purl/400027 this] for more about the FBI's decision to digitalize fingerprint files and the process behind it.
* ''[expand upon Fermat's optimization? Use the PDF?]''
* '''Fermet's Theorem =''' If a real-valued function, &lt;math&gt;f(x)&lt;/math&gt;, is differentiable&lt;ref&gt;function has a well-defined, smooth slope at every single point&lt;/ref&gt; in an interval &lt;math&gt;(a, b)&lt;/math&gt; and &lt;math&gt;f(x)&lt;/math&gt; has a maximum OR minimum at &lt;math&gt;c&lt;/math&gt; ∈ &lt;math&gt;(a, b)&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;f'(c)&lt;/math&gt; = &lt;math display=&quot;inline&quot;&gt;0&lt;/math&gt;&lt;ref&gt;{{Cite web|url=https://old.maa.org/press/periodicals/convergence/fermat-s-method-for-finding-maxima-and-minima-a-mini-primary-source-project-for-calculus-1-students|title=Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students {{!}} Mathematical Association of America|website=old.maa.org|access-date=2026-05-21}}&lt;/ref&gt;.
** Explanation of ∈: essentially &quot;belongs to/inside/a member of.&quot; For example, &lt;math&gt;c&lt;/math&gt; ∈ &lt;math&gt;(a, b)&lt;/math&gt; → &quot;the number c&lt;math&gt;&lt;/math&gt; is inside the interval between &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt;&quot;.

=== 5/23/2026 [Logarithms] ===
''[insert logarithms introduction/lesson]''

log(''a'' x ''b'') = log ''a'' + log ''b''

Multiply two numbers together, take the log = answer is the SUM of their individual logs. Logarithms are like an &quot;undo&quot; tool. They &quot;undo&quot; the mathematical operations done by exponential functions, and the relationship between logarithms and exponential functions is reciprocal. 

* ''e''  = 2.71828... similar to π in circles&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=136}}&lt;/ref&gt;. See [https://simple.wikipedia.org/wiki/E_(mathematical_constant) e (mathematical constant)] (simple-wiki) &amp; [[w:Natural logarithm]] (wikipedia). The rate of change of ''e''&lt;sup&gt;x&lt;/sup&gt; is ''e''&lt;sup&gt;x&lt;/sup&gt;. The rate of exponential growth is proportional to the function's current level&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=137}}&lt;/ref&gt;. An example to illustrate this is the following: as a microphone picks up a noise that increases in volume (perhaps the source of the sound is moving closer to the microphone), the loudspeaker amplifies the noise at a constant, exponential rate ''in proportional'' (NOT equal) to the noise it is picking up through the microphone.

=== 5/27/2026 [Derivatives] ===
[[File:2020-03-25 00 08 15 A Five Cheese Pizza Hot Pocket after being heated in the Franklin Farm section of Oak Hill, Fairfax County, Virginia.jpg|thumb|When looking at how many ''more'' calories I will consume per infinitesimally small bite of the hot pocket, we are assessing the derivative of the hot pocket's calories. 

Yes, this may not be practical, but hopefully bringing food into the 'equation' will help you understand the concept of derivatives better.]]

* What is the definition of a '''derivative'''? Essentially the rate of change: ''dy/dx''. An example of a derivative is [[PlanetPhysics/Acceleration|acceleration]]. Another example of a derivative is the following question: how many calories will I consume per bite of a hot pocket (each bite being infinitesimally small)?

The question posed by the book is as follows: ''how do we define the slope when the slope keeps changing?''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=143}}&lt;/ref&gt;

Shifting our mindset from [[Speak Math Now!|algebra]]: In calculus, the rate of change is ''not'' constant, as the IV changes (and is therefore regarded as a '''function'''). We go from Δy/Δx [set rate of change] → ''dy/dx'' [infinitesimally tiny, varied changes]. 

So instead of thinking of the hourly rate for a cashier as a set number (let's say $16/hr), we should think of the $16/hr as a ''constant'' function. This is going to pay off in calculus as we deal with rates of changes that are not always 'set in stone', or constant. For example, measuring a horse's total speed in a [[w:Horse_racing|horse race]] is not going to be a constant, set number - it will be a function with a constantly changing rate. For this specific example:

* '''x''' = time
* '''y''' = speed
* '''dy/dx''' = rate of change of horse's speed with respect to time (think of it as: &quot;rate of change of [y] in respect to [x]&quot;).

=== 6/6/2026 [Definied Integrals &amp; Area Function] ===
''background info...''

A '''definite''' '''integral''', in calculus, is the generalized area under the curved function&lt;ref&gt;{{Cite web|url=https://openstax.org/books/calculus-volume-1/pages/5-2-the-definite-integral|title=5.2 The Definite Integral - Calculus Volume 1 {{!}} OpenStax|last=Strang|first=Gilbert|last2=Herman|first2=Edwin “Jed”|date=2016-03-30|website=openstax.org|language=English|access-date=2026-06-06}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/definite-integral-as-the-limit-of-a-riemann-sum|title=Khan Academy|website=www.khanacademy.org|language=en|access-date=2026-06-06}}&lt;/ref&gt;.

* '''Area function''' (calculus) - Accumulated area under a curved line on an ''xy'' graph from point ''a'' to point ''x'' [upper bound that can be moved as opposed to point ''b'']. [https://www.youtube.com/watch?v=6WfetaviTrQ YT video].

TO-DO [6/6/2026]: provide example problem of area function modeling of YT video, do the work, take a pic and upload.

== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]
'''[[w:Pierre_de_Fermat|Pierre de Fermat]]'''

* [https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf ''Fermat’s Method for Finding Maxima and Minima'']- Kenneth M Monks (2023)
'''Other'''

* [[w:Glossary_of_mathematical_symbols|Glossary of mathematical symbols]]

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
{{reflist}}
[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
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      <text bytes="14869" sha1="4g4khq4jucr1zcpexxplkudl0ygxu5g" xml:space="preserve">{{mathematics}}'''&lt;u&gt;Book&lt;/u&gt;''': ''Infinite Powers'' by Steven Strogatz (ISBN#: 1328879984){{tertiary}}
{{Notes}}
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{{contrib-creator|[[User:Atcovi|Atcovi]]}}
== Notes ==
[[File:Parts of Parabola.svg|thumb|A diagram of a parabola.]]

=== 4/11/2026 (Archimedes and the method of exhaustion) ===
* Archimedes and figuring out the ''quadratic'' (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a [[parabola]] to equal one big triangle (&lt;math display=&quot;inline&quot;&gt;=1&lt;/math&gt;) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: &lt;math display=&quot;inline&quot;&gt;1&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/16&lt;/math&gt; + &lt;math display=&quot;inline&quot;&gt;1/64&lt;/math&gt; ← geometric series.

^each term is &lt;math display=&quot;inline&quot;&gt;1/4&lt;/math&gt; of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do. 

Archimedes proved that &lt;math display=&quot;inline&quot;&gt;a = 4/3&lt;/math&gt; through a '''double reductio ad absurdum'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=36}}&lt;/ref&gt; using the '''method of exhaustion''', an analytical way of finding a result&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=102}}&lt;/ref&gt;.

=== 5/2/2026 (Johannes Kepler) ===

==== [[w:Johannes_Kepler|Johannes Kepler]] ====

# '''[[w:Elliptic orbit|Elliptical orbits]]'''
#*'''Ellipse''': Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit. 
# '''[https://www.socratica.com/pages/keplers-second-law-of-motion Equal Areas in Equal Times]'''
#*'''Formula''': Time (P&lt;sub&gt;1&lt;/sub&gt; → P&lt;sub&gt;2&lt;/sub&gt;) = Time (P&lt;sub&gt;3&lt;/sub&gt; → P&lt;sub&gt;4&lt;/sub&gt;) [their sectors have equal areas]
# '''Third Law and the Sacred Frenzy'''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=84}}&lt;/ref&gt;
#*&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt;&lt;sup&gt;3&lt;/sup&gt;
#**&lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; = how long it takes for a planet to go around the sun just once.
#**&lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; = avg. of the planet's nearest and farthest distance from the sun.

=== 5/14/2026 (Calculus definitions, introduction to adequality) ===
* '''[[w:Differential_calculus|Differential calculus]]:''' cuts complicated problems into infinitely many simpler pieces. Ex, derivatives. 
* '''[[w:Integral_calculus|Integral calculus]]''': puts the pieces back together again to solve the original problem. Ex, integrals. 
[[File:Tangent function animation.gif|thumb|The derivative at different points of a differentiable function. In this case, the derivative is equal to &lt;math&gt;\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right)&lt;/math&gt;.&lt;ref&gt;{{Cite journal|date=2026-04-13|title=Derivative|url=https://en.wikipedia.org/w/index.php?title=Derivative&amp;oldid=1348562692|journal=Wikipedia|language=en}}&lt;/ref&gt;]]
[[File:Cartesian-coordinate-system.svg|thumb|This is known as a ''Cartesian coordinate system''.|left]]
* '''[[w:Analytical_geometry|Analytical geometry]]''': Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. &quot;Analysis&quot; in analytic geometry is meant to be understood as a way of ''figuring out'' the results rather than proving the results&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=101}}&lt;/ref&gt;. 
==== Adequality ====
''See pages 103 to 107, which provide a breakdown of [[w:Pierre_de_Fermat|Pierre de Fermat]] and his concept of adequality.''

Pierre de Fermat's concept of adequality (meaning ''approximate equality''&lt;ref&gt;{{Cite journal|date=2024-09-18|title=Number Theory: An Approach Through History from Hammurapi to Legendre|url=https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&amp;oldid=1246411217|journal=Wikipedia|language=en}}&lt;/ref&gt;) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] ''a'' and ''b'' at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=106}}&lt;/ref&gt;.

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus. 
=== 5/16/2026 (continuation of Fermat's adequality) ===
[[File:Week 9 Fermat and Adequality Proto-Calculus Notes - Part 1.jpg|thumb|438x438px|'''Figure 1.''' Written statements [in all caps] are as follows (from the top-down): 1. WHAT IS THE MAXIMUM VALUE? 2. TWO NEARBY X-VALUES, X&lt;sub&gt;1&lt;/sub&gt; AND X&lt;sub&gt;2&lt;/sub&gt;, PRODUCE ALMOST THE SAME OUTPUT; l = left side, r = right side in the hill diagram]]
==== What does b - (x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;) = 0 represent? ====
b = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt;

Reference the hill diagram in '''Figure 1''' (you may have to open the file and zoom in). X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; represent two nearby points on both sides of the &quot;hill&quot; which both produce almost the same output.  

For both of the values, adding both X&lt;sub&gt;1&lt;/sub&gt; and X&lt;sub&gt;2&lt;/sub&gt; would equal &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt; (the total length). B = x&lt;sub&gt;1&lt;/sub&gt; + x&lt;sub&gt;2&lt;/sub&gt; would come out to B = 2x, with '''x = b/2''' (where the maximum is). This is the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would ideally give the highest value for &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (see below).
==== Purpose of bx - x&lt;sup&gt;2&lt;/sup&gt; = c? ====
What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&amp;t [minute 4:17] for context/writing of this equation): &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;?

If we take a line (total = &lt;math display=&quot;inline&quot;&gt;b&lt;/math&gt;), and make a cut at some point in the line (and designate the cut 'mark' as &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;), how could we figure out &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; (output produced by the equation, &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt; = &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt;)? 

&lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; represents a portion of the line, while &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; represents the remaining portion of the line. The product of both &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; and &lt;math display=&quot;inline&quot;&gt;b - x&lt;/math&gt; is &lt;math display=&quot;inline&quot;&gt;bx&lt;/math&gt; - &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;&lt;sup&gt;2&lt;/sup&gt;. The goal is to find the value of &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; that would produce the highest &lt;math display=&quot;inline&quot;&gt;c&lt;/math&gt; value.

=== 5/20/2026 [Fermet's Theorem] ===

* Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read [https://www.osti.gov/servlets/purl/400027 this] for more about the FBI's decision to digitalize fingerprint files and the process behind it.
* ''[expand upon Fermat's optimization? Use the PDF?]''
* '''Fermet's Theorem =''' If a real-valued function, &lt;math&gt;f(x)&lt;/math&gt;, is differentiable&lt;ref&gt;function has a well-defined, smooth slope at every single point&lt;/ref&gt; in an interval &lt;math&gt;(a, b)&lt;/math&gt; and &lt;math&gt;f(x)&lt;/math&gt; has a maximum OR minimum at &lt;math&gt;c&lt;/math&gt; ∈ &lt;math&gt;(a, b)&lt;/math&gt;, then &lt;math display=&quot;inline&quot;&gt;f'(c)&lt;/math&gt; = &lt;math display=&quot;inline&quot;&gt;0&lt;/math&gt;&lt;ref&gt;{{Cite web|url=https://old.maa.org/press/periodicals/convergence/fermat-s-method-for-finding-maxima-and-minima-a-mini-primary-source-project-for-calculus-1-students|title=Fermat’s Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students {{!}} Mathematical Association of America|website=old.maa.org|access-date=2026-05-21}}&lt;/ref&gt;.
** Explanation of ∈: essentially &quot;belongs to/inside/a member of.&quot; For example, &lt;math&gt;c&lt;/math&gt; ∈ &lt;math&gt;(a, b)&lt;/math&gt; → &quot;the number c&lt;math&gt;&lt;/math&gt; is inside the interval between &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt;&quot;.

=== 5/23/2026 [Logarithms] ===
''[insert logarithms introduction/lesson]''

log(''a'' x ''b'') = log ''a'' + log ''b''

Multiply two numbers together, take the log = answer is the SUM of their individual logs. Logarithms are like an &quot;undo&quot; tool. They &quot;undo&quot; the mathematical operations done by exponential functions, and the relationship between logarithms and exponential functions is reciprocal. 

* ''e''  = 2.71828... similar to π in circles&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=136}}&lt;/ref&gt;. See [https://simple.wikipedia.org/wiki/E_(mathematical_constant) e (mathematical constant)] (simple-wiki) &amp; [[w:Natural logarithm]] (wikipedia). The rate of change of ''e''&lt;sup&gt;x&lt;/sup&gt; is ''e''&lt;sup&gt;x&lt;/sup&gt;. The rate of exponential growth is proportional to the function's current level&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=137}}&lt;/ref&gt;. An example to illustrate this is the following: as a microphone picks up a noise that increases in volume (perhaps the source of the sound is moving closer to the microphone), the loudspeaker amplifies the noise at a constant, exponential rate ''in proportional'' (NOT equal) to the noise it is picking up through the microphone.

=== 5/27/2026 [Derivatives] ===
[[File:2020-03-25 00 08 15 A Five Cheese Pizza Hot Pocket after being heated in the Franklin Farm section of Oak Hill, Fairfax County, Virginia.jpg|thumb|When looking at how many ''more'' calories I will consume per infinitesimally small bite of the hot pocket, we are assessing the derivative of the hot pocket's calories. 

Yes, this may not be practical, but hopefully bringing food into the 'equation' will help you understand the concept of derivatives better.]]

* What is the definition of a '''derivative'''? Essentially the rate of change: ''dy/dx''. An example of a derivative is [[PlanetPhysics/Acceleration|acceleration]]. Another example of a derivative is the following question: how many calories will I consume per bite of a hot pocket (each bite being infinitesimally small)?

The question posed by the book is as follows: ''how do we define the slope when the slope keeps changing?''&lt;ref&gt;{{Cite book|title=Infinite powers: how calculus reveals the secrets of the universe|last=Strogatz|first=Steven|date=2020|publisher=Mariner Books ; Houghton Mifflin Harcourt|isbn=978-1-328-87998-1|edition=First Mariner books edition|location=Boston New York|page=143}}&lt;/ref&gt;

Shifting our mindset from [[Speak Math Now!|algebra]]: In calculus, the rate of change is ''not'' constant, as the IV changes (and is therefore regarded as a '''function'''). We go from Δy/Δx [set rate of change] → ''dy/dx'' [infinitesimally tiny, varied changes]. 

So instead of thinking of the hourly rate for a cashier as a set number (let's say $16/hr), we should think of the $16/hr as a ''constant'' function. This is going to pay off in calculus as we deal with rates of changes that are not always 'set in stone', or constant. For example, measuring a horse's total speed in a [[w:Horse_racing|horse race]] is not going to be a constant, set number - it will be a function with a constantly changing rate. For this specific example:

* '''x''' = time
* '''y''' = speed
* '''dy/dx''' = rate of change of horse's speed with respect to time (think of it as: &quot;rate of change of [y] in respect to [x]&quot;).

=== 6/6/2026 [Definite Integrals &amp; Area Function] ===
''background info...''

A '''definite''' '''integral''', in calculus, is the generalized area under the curved function&lt;ref&gt;{{Cite web|url=https://openstax.org/books/calculus-volume-1/pages/5-2-the-definite-integral|title=5.2 The Definite Integral - Calculus Volume 1 {{!}} OpenStax|last=Strang|first=Gilbert|last2=Herman|first2=Edwin “Jed”|date=2016-03-30|website=openstax.org|language=English|access-date=2026-06-06}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/definite-integral-as-the-limit-of-a-riemann-sum|title=Khan Academy|website=www.khanacademy.org|language=en|access-date=2026-06-06}}&lt;/ref&gt;.

* '''Area function''' (calculus) - Accumulated area under a curved line on an ''xy'' graph from point ''a'' to point ''x'' [upper bound that can be moved as opposed to point ''b'']. [https://www.youtube.com/watch?v=6WfetaviTrQ YT video].

TO-DO [6/6/2026]: provide example problem of area function modeling of YT video, do the work, take a pic and upload.

== Wikipedia/Study Links ==
[[w:Archimedes|'''Archimedes''']]
* [[w:Approximations_of_pi|approximations of pi]]
* quadrature (computation of area) of a parabolic segment
* [[w:Archimedes_Palimpsest|''Archimedes Palimpsest'']]
* [https://math.nyu.edu/Archimedes/Lever/LeverLaw.html Archimedes' Law of the Lever]
'''[[w:Pierre_de_Fermat|Pierre de Fermat]]'''

* [https://old.maa.org/sites/default/files/images/upload_library/46/Barnett_TRIUMPHS_MiniPSPs/MiniPSP_FermatsMethod_2023_02_20.pdf ''Fermat’s Method for Finding Maxima and Minima'']- Kenneth M Monks (2023)
'''Other'''

* [[w:Glossary_of_mathematical_symbols|Glossary of mathematical symbols]]

== See Also ==
* [[User:Addemf/sandbox/Who Invented Calculus?]]

== References/Sources ==
{{reflist}}
[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
      <sha1>4g4khq4jucr1zcpexxplkudl0ygxu5g</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Original research and scholarly standards</title>
    <ns>4</ns>
    <id>329789</id>
    <revision>
      <id>2813303</id>
      <parentid>2812818</parentid>
      <timestamp>2026-06-06T14:37:45Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>proposed process</comment>
      <origin>2813303</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5275" sha1="1gwtv3xa60v4jtx4aj61snstsf6lcb7" xml:space="preserve">{{merge|Wikiversity:Original research}}
{{proposal}}
{{info|This is a rough draft}}
''Purpose'': Explanation of acceptable/unacceptable original research, bridging the gap of all Wikiversity pages that detail research, research standards, and commitment to academic learning and growth. 

Wikiversity, as a part of the Wikimedia Foundation, has a commitment to [[Wikiversity:Learning by doing|active]] [[Wikiversity:Learning|learning]] that is in adherence to proper, high-quality research standards. This includes adhering to [[Wikiversity:Research guidelines#Ethics|scholarly ethics]] when conducting and presenting research, which includes:

* subjecting research to [[Wikiversity:Peer review|peer review]] and independent verification, whichever and wherever its applicable.
* not violating established ethical guidelines
* honesty, including transparency, listing objectives, [[Wikiversity:Verifiability|citing reliable sources]], and making note of any [[Wikiversity:Disclosures|notable disclosures]]. See [[Wikiversity:Research_process#What_are_research_ethics?]] for more research ethics that editors should strive for. 

Wikiversity, as part of its [[Wikiversity:What is Wikiversity?|aims]], encourages &quot;learning by doing&quot;, or [[Wikiversity:Developing Wikiversity through action research|through active research]]. Unlike its counterpart, [[Wikiversity:Wikipedia|Wikipedia]], Wikiversity allows [[Wikiversity:Original research|original research.]] Briefly, original research examples include:

# Conducting an experiment testing the waterproofability of three brand wallets.
# [[Help:Lab reports|Lab reports]] conducted in a scientific venue, such as [[Help:Assignment|class assignments]]. See examples of lab reports in this category: [[:Category:Lab reports]]. 
# Academic essays as part of homework assignments. See examples of essays here: [[:Category:Essays]].

While Wikiversity enjoys the benefit of flexibility, Wikiversity strives to avoid content that promotes severe deviations from mainstream science, and speculative theories that present themselves as established science without proper contextualization/disclosure (such as the promotion of [[wikipedia:Fringe_theory|fringe theories]] or [[wikipedia:Pseudoscience|pseudoscience]]). Such content harms not only the reputation of Wikiversity but also the ability of viewers and collaborators to properly engage and foster a learning environment&lt;ref&gt;''Using Wikiversity as an academic discussion forum may help share ideas that may promote research and learning.'' - [[Wikiversity:What_is_Wikiversity?#Wikiversity_for_sharing_materials,_ideas,_community]]&lt;/ref&gt;. 

Original research '''must''' be presented in a way that readers can understand that the authors are presenting new ideas that are in accordance with scientific practices, including honest disclosures (including NPOV), distinction between speculation and established knowledge, and invitation of peer review (through the [[Template:To be peer reviewed|to be peer reviewed]] template; though this does not serve as a &quot;green light&quot; for fringe research)&lt;ref&gt;would require changing [[Wikiversity:Original research]] requirements.&lt;/ref&gt;. 

Research that deviates from standard scientific practices includes presenting scientific theories without reliable sources backing them up, using or manipulating scientific terminology, making extreme claims, or failing to provide a clear learning structure revolving around [[Wikiversity:Research collaboration|collaborative learning]]. Pages that fail to meet these requirements may be moved to userspace/draft, require cleanup notices, or be heavily rewritten as they do not meet Wikiversity's [[Wikiversity:Scope|scope]]&lt;ref&gt;''Wikiversity offers a collaborative environment for the creation, sharing, and discussion of [[open educational resources]], [[open research]] and [[open academia]].'' ([[Wikiversity:Scope]]) - derived from this.&lt;/ref&gt;. 

The essential rule is to be honest with your readers and to contextualize the learning resource you are presenting so viewers can extract as much learning value from your resource as possible. 

== Guidelines/Checklist ==
== [Proposed] Process ==
#All original research must first be submitted under the &quot;Draft&quot; namespace.
#Author of the page must submit it for mainspace consideration at ''[page name?]''
#The [[Wikiversity:Review board|Review board]] (Wikiversity bureaucrats) must reach a consensus to approve certain pages in accordance with Wikiversity's original research policy.
== See also ==
'''Wikiversity Space Links'''

* [[Wikiversity:Scope]]

* [[Wikiversity:Original research]] (proposed policy)

* [[Wikiversity:Research]]
* [[Wikiversity:Research process]]
** [[Wikiversity:Research_process#What_are_research_ethics?]]
* [[Wikiversity:Scholarly ethics]]
* [[Wikiversity:Peer review]]
** [[Wikiversity:Peer review verification]]
* [[Wikiversity:Scope of research]]
* [[Wikiversity:POV]]
* [[Wikiversity:What is Wikiversity?]]
* [[Wikiversity:What Wikiversity is not]]

'''Pseudoscience/Fringe Theories'''
* [[wikipedia:Pseudoscience]]
* [[wikipedia:Wikipedia:Fringe_theories_for_dummies|Wikipedia:Fringe theories for dummies]]
* [[Wikipedia:Wikipedia:Fringe_theories#Pseudoscience]]

== Notes ==
&lt;references /&gt;
[[Category:Atcovi's Work]]</text>
      <sha1>1gwtv3xa60v4jtx4aj61snstsf6lcb7</sha1>
    </revision>
    <revision>
      <id>2813304</id>
      <parentid>2813303</parentid>
      <timestamp>2026-06-06T14:38:04Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* [Proposed] Process */ #4</comment>
      <origin>2813304</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5334" sha1="jxj7bej0j9ciokelgtppfm7inmu0lp5" xml:space="preserve">{{merge|Wikiversity:Original research}}
{{proposal}}
{{info|This is a rough draft}}
''Purpose'': Explanation of acceptable/unacceptable original research, bridging the gap of all Wikiversity pages that detail research, research standards, and commitment to academic learning and growth. 

Wikiversity, as a part of the Wikimedia Foundation, has a commitment to [[Wikiversity:Learning by doing|active]] [[Wikiversity:Learning|learning]] that is in adherence to proper, high-quality research standards. This includes adhering to [[Wikiversity:Research guidelines#Ethics|scholarly ethics]] when conducting and presenting research, which includes:

* subjecting research to [[Wikiversity:Peer review|peer review]] and independent verification, whichever and wherever its applicable.
* not violating established ethical guidelines
* honesty, including transparency, listing objectives, [[Wikiversity:Verifiability|citing reliable sources]], and making note of any [[Wikiversity:Disclosures|notable disclosures]]. See [[Wikiversity:Research_process#What_are_research_ethics?]] for more research ethics that editors should strive for. 

Wikiversity, as part of its [[Wikiversity:What is Wikiversity?|aims]], encourages &quot;learning by doing&quot;, or [[Wikiversity:Developing Wikiversity through action research|through active research]]. Unlike its counterpart, [[Wikiversity:Wikipedia|Wikipedia]], Wikiversity allows [[Wikiversity:Original research|original research.]] Briefly, original research examples include:

# Conducting an experiment testing the waterproofability of three brand wallets.
# [[Help:Lab reports|Lab reports]] conducted in a scientific venue, such as [[Help:Assignment|class assignments]]. See examples of lab reports in this category: [[:Category:Lab reports]]. 
# Academic essays as part of homework assignments. See examples of essays here: [[:Category:Essays]].

While Wikiversity enjoys the benefit of flexibility, Wikiversity strives to avoid content that promotes severe deviations from mainstream science, and speculative theories that present themselves as established science without proper contextualization/disclosure (such as the promotion of [[wikipedia:Fringe_theory|fringe theories]] or [[wikipedia:Pseudoscience|pseudoscience]]). Such content harms not only the reputation of Wikiversity but also the ability of viewers and collaborators to properly engage and foster a learning environment&lt;ref&gt;''Using Wikiversity as an academic discussion forum may help share ideas that may promote research and learning.'' - [[Wikiversity:What_is_Wikiversity?#Wikiversity_for_sharing_materials,_ideas,_community]]&lt;/ref&gt;. 

Original research '''must''' be presented in a way that readers can understand that the authors are presenting new ideas that are in accordance with scientific practices, including honest disclosures (including NPOV), distinction between speculation and established knowledge, and invitation of peer review (through the [[Template:To be peer reviewed|to be peer reviewed]] template; though this does not serve as a &quot;green light&quot; for fringe research)&lt;ref&gt;would require changing [[Wikiversity:Original research]] requirements.&lt;/ref&gt;. 

Research that deviates from standard scientific practices includes presenting scientific theories without reliable sources backing them up, using or manipulating scientific terminology, making extreme claims, or failing to provide a clear learning structure revolving around [[Wikiversity:Research collaboration|collaborative learning]]. Pages that fail to meet these requirements may be moved to userspace/draft, require cleanup notices, or be heavily rewritten as they do not meet Wikiversity's [[Wikiversity:Scope|scope]]&lt;ref&gt;''Wikiversity offers a collaborative environment for the creation, sharing, and discussion of [[open educational resources]], [[open research]] and [[open academia]].'' ([[Wikiversity:Scope]]) - derived from this.&lt;/ref&gt;. 

The essential rule is to be honest with your readers and to contextualize the learning resource you are presenting so viewers can extract as much learning value from your resource as possible. 

== Guidelines/Checklist ==
== [Proposed] Process ==
#All original research must first be submitted under the &quot;Draft&quot; namespace.
#Author of the page must submit it for mainspace consideration at ''[page name?]''
#The [[Wikiversity:Review board|Review board]] (Wikiversity bureaucrats) must reach a consensus to approve certain pages in accordance with Wikiversity's original research policy.
#If successful, the page may be moved into the mainspace.

== See also ==
'''Wikiversity Space Links'''

* [[Wikiversity:Scope]]

* [[Wikiversity:Original research]] (proposed policy)

* [[Wikiversity:Research]]
* [[Wikiversity:Research process]]
** [[Wikiversity:Research_process#What_are_research_ethics?]]
* [[Wikiversity:Scholarly ethics]]
* [[Wikiversity:Peer review]]
** [[Wikiversity:Peer review verification]]
* [[Wikiversity:Scope of research]]
* [[Wikiversity:POV]]
* [[Wikiversity:What is Wikiversity?]]
* [[Wikiversity:What Wikiversity is not]]

'''Pseudoscience/Fringe Theories'''
* [[wikipedia:Pseudoscience]]
* [[wikipedia:Wikipedia:Fringe_theories_for_dummies|Wikipedia:Fringe theories for dummies]]
* [[Wikipedia:Wikipedia:Fringe_theories#Pseudoscience]]

== Notes ==
&lt;references /&gt;
[[Category:Atcovi's Work]]</text>
      <sha1>jxj7bej0j9ciokelgtppfm7inmu0lp5</sha1>
    </revision>
    <revision>
      <id>2813305</id>
      <parentid>2813304</parentid>
      <timestamp>2026-06-06T14:38:14Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* [Proposed] Process */ implied</comment>
      <origin>2813305</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5323" sha1="rk94iyu57rz5j866ii98eq27ihi88l2" xml:space="preserve">{{merge|Wikiversity:Original research}}
{{proposal}}
{{info|This is a rough draft}}
''Purpose'': Explanation of acceptable/unacceptable original research, bridging the gap of all Wikiversity pages that detail research, research standards, and commitment to academic learning and growth. 

Wikiversity, as a part of the Wikimedia Foundation, has a commitment to [[Wikiversity:Learning by doing|active]] [[Wikiversity:Learning|learning]] that is in adherence to proper, high-quality research standards. This includes adhering to [[Wikiversity:Research guidelines#Ethics|scholarly ethics]] when conducting and presenting research, which includes:

* subjecting research to [[Wikiversity:Peer review|peer review]] and independent verification, whichever and wherever its applicable.
* not violating established ethical guidelines
* honesty, including transparency, listing objectives, [[Wikiversity:Verifiability|citing reliable sources]], and making note of any [[Wikiversity:Disclosures|notable disclosures]]. See [[Wikiversity:Research_process#What_are_research_ethics?]] for more research ethics that editors should strive for. 

Wikiversity, as part of its [[Wikiversity:What is Wikiversity?|aims]], encourages &quot;learning by doing&quot;, or [[Wikiversity:Developing Wikiversity through action research|through active research]]. Unlike its counterpart, [[Wikiversity:Wikipedia|Wikipedia]], Wikiversity allows [[Wikiversity:Original research|original research.]] Briefly, original research examples include:

# Conducting an experiment testing the waterproofability of three brand wallets.
# [[Help:Lab reports|Lab reports]] conducted in a scientific venue, such as [[Help:Assignment|class assignments]]. See examples of lab reports in this category: [[:Category:Lab reports]]. 
# Academic essays as part of homework assignments. See examples of essays here: [[:Category:Essays]].

While Wikiversity enjoys the benefit of flexibility, Wikiversity strives to avoid content that promotes severe deviations from mainstream science, and speculative theories that present themselves as established science without proper contextualization/disclosure (such as the promotion of [[wikipedia:Fringe_theory|fringe theories]] or [[wikipedia:Pseudoscience|pseudoscience]]). Such content harms not only the reputation of Wikiversity but also the ability of viewers and collaborators to properly engage and foster a learning environment&lt;ref&gt;''Using Wikiversity as an academic discussion forum may help share ideas that may promote research and learning.'' - [[Wikiversity:What_is_Wikiversity?#Wikiversity_for_sharing_materials,_ideas,_community]]&lt;/ref&gt;. 

Original research '''must''' be presented in a way that readers can understand that the authors are presenting new ideas that are in accordance with scientific practices, including honest disclosures (including NPOV), distinction between speculation and established knowledge, and invitation of peer review (through the [[Template:To be peer reviewed|to be peer reviewed]] template; though this does not serve as a &quot;green light&quot; for fringe research)&lt;ref&gt;would require changing [[Wikiversity:Original research]] requirements.&lt;/ref&gt;. 

Research that deviates from standard scientific practices includes presenting scientific theories without reliable sources backing them up, using or manipulating scientific terminology, making extreme claims, or failing to provide a clear learning structure revolving around [[Wikiversity:Research collaboration|collaborative learning]]. Pages that fail to meet these requirements may be moved to userspace/draft, require cleanup notices, or be heavily rewritten as they do not meet Wikiversity's [[Wikiversity:Scope|scope]]&lt;ref&gt;''Wikiversity offers a collaborative environment for the creation, sharing, and discussion of [[open educational resources]], [[open research]] and [[open academia]].'' ([[Wikiversity:Scope]]) - derived from this.&lt;/ref&gt;. 

The essential rule is to be honest with your readers and to contextualize the learning resource you are presenting so viewers can extract as much learning value from your resource as possible. 

== Guidelines/Checklist ==
== Process ==
#All original research must first be submitted under the &quot;Draft&quot; namespace.
#Author of the page must submit it for mainspace consideration at ''[page name?]''
#The [[Wikiversity:Review board|Review board]] (Wikiversity bureaucrats) must reach a consensus to approve certain pages in accordance with Wikiversity's original research policy.
#If successful, the page may be moved into the mainspace.

== See also ==
'''Wikiversity Space Links'''

* [[Wikiversity:Scope]]

* [[Wikiversity:Original research]] (proposed policy)

* [[Wikiversity:Research]]
* [[Wikiversity:Research process]]
** [[Wikiversity:Research_process#What_are_research_ethics?]]
* [[Wikiversity:Scholarly ethics]]
* [[Wikiversity:Peer review]]
** [[Wikiversity:Peer review verification]]
* [[Wikiversity:Scope of research]]
* [[Wikiversity:POV]]
* [[Wikiversity:What is Wikiversity?]]
* [[Wikiversity:What Wikiversity is not]]

'''Pseudoscience/Fringe Theories'''
* [[wikipedia:Pseudoscience]]
* [[wikipedia:Wikipedia:Fringe_theories_for_dummies|Wikipedia:Fringe theories for dummies]]
* [[Wikipedia:Wikipedia:Fringe_theories#Pseudoscience]]

== Notes ==
&lt;references /&gt;
[[Category:Atcovi's Work]]</text>
      <sha1>rk94iyu57rz5j866ii98eq27ihi88l2</sha1>
    </revision>
  </page>
  <page>
    <title>Social Victorians/Irish Aristocracy</title>
    <ns>0</ns>
    <id>329829</id>
    <revision>
      <id>2813346</id>
      <parentid>2812976</parentid>
      <timestamp>2026-06-06T22:41:40Z</timestamp>
      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
      </contributor>
      <origin>2813346</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="40470" sha1="ksnbj11xn7t5a0q5n461o9pf5u935l6" xml:space="preserve">= The Irish Aristocracy at the End of the 19th Century =

== The Irish Peerage ==
Minus the people who attended the ball, which are in [[Social Victorians/Irish Aristocracy#Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball|this section, below]].

=== Dukes and Duchesses ===

==== Duke of Leinster ====
Irish peerage

* Gerald FitzGerald, 5th Duke of Leinster (16 August 1851 – 1 December 1893)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p1207.htm#i12063|title=Gerald FitzGerald, 5th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}&lt;/ref&gt;
* Maurice FitzGerald, 6th Duke of Leinster, 6 years old when he succeeded to the dukedom&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2767.htm#i27667|title=Maurice FitzGerald, 6th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}&lt;/ref&gt;

* Subsidiary Titles
# Marquess of Kildare (Irish peerage), did not attend the ball.
# Earl of Kildare (Irish peerage), did not attend the ball.
# Earl of Offaly (Irish peerage)
# Viscount Leinster of Taplow (GB peerage)
# Baron Offaly (Irish peerage)
# Baron Kildare of Kildare (UK peerage)

=== Marquesses and Marchionesses ===

==== Marquess Conyngham&lt;ref&gt;{{Cite journal|date=2026-01-13|title=Marquess Conyngham|url=https://en.wikipedia.org/w/index.php?title=Marquess_Conyngham&amp;oldid=1332742873|journal=Wikipedia|language=en}}&lt;/ref&gt; ====

* Did not attend the ball but did attend a number of social events about this time.
* Pronounced &quot;''Cunn''ingum.&quot;&lt;ref&gt;{{Cite journal|date=2026-01-13|title=Marquess Conyngham|url=https://en.wikipedia.org/w/index.php?title=Marquess_Conyngham&amp;oldid=1332742873|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Henry Francis Conyngham, 4th Marquess Conyngham (1857–1897)&lt;ref&gt;&quot;Henry Francis Conyngham, 4th Marquess Conyngham.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 7198

https://www.thepeerage.com/p7199.htm#i71982.&lt;/ref&gt;
* Victor George Henry Francis Conyngham, 5th Marquess Conyngham (1883–1918)&lt;ref&gt;&quot;Victor George Henry Francis Conyngham, 5th Marquess Conyngham.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 7198 https://www.thepeerage.com/p7199.htm#i71983.&lt;/ref&gt;

* Subsidiary Titles
** Earl of Conyngham
** Viscount Conyngham
** Viscount Mount Charles

==== Marquess of Donegall ====

* Did not attend the ball.
* Subsidiary Titles
** Earl of Donegall, did not attend the ball.
** Viscount Chichester — did not attend the ball; some Chichesters attended social events at about this time.

==== Marquess and Marchioness of Downshire ====
* Arthur Wills John Wellington Trumbull Blundell Hill, 6th Marquess of Downshire (2 July 1871 – 29 May 1918) in 1893 married Katherine Mary (&quot;Kitty&quot;) Hare (1872–1959)&lt;ref&gt;{{Cite journal|date=2025-02-10|title=Arthur Hill, 6th Marquess of Downshire|url=https://en.wikipedia.org/w/index.php?title=Arthur_Hill,_6th_Marquess_of_Downshire&amp;oldid=1274976272|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Did not attend the ball.
* Subsidiary Titles
** Earl of Hillsborough, did not attend the ball, also not at any social events described so far.
** Viscount Kilwarlin — 6th, Arthur Wills John Wellington Trumbull Hill (31 March 1874 – 29 May 1918)&lt;ref&gt;&quot;Arthur Wills John Wellington Trumbull '''Hill''', 6th Marquess of Downshire.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page #3810

https://www.thepeerage.com/p3811.htm#i38104.&lt;/ref&gt;

==== Marquess of Ely ====

* Did not attend the ball, but members of the Loftus family attended a number of social events at about this time.
* 4th Marquess: John Henry Wellington Graham Loftus (15 July 1857 – 3 April 1889)&lt;ref&gt;&quot;John Henry Wellington Graham Loftus, 4th Marquess of Ely.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 8545 https://www.thepeerage.com/p8545.htm#i85450.&lt;/ref&gt;
* 5th Marquess: John Henry Loftus (3 April 1889 – 18 December 1925)&lt;ref&gt;&quot;John Henry Loftus, 5th Marquess of Ely.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 8546 https://www.thepeerage.com/p8546.htm#i85459.&lt;/ref&gt;
* Subsidiary Titles
** Earl of Ely — did not attend the ball.
** Viscount Loftus

==== [[Social Victorians/People/Bective|Marquess and Marchioness of Headfort]] ====
* Did not attend the ball, but a number of people in this family attended many social events at about this time.
* Subsidiary Titles
** [[Social Victorians/People/Bective|Earl of Bective]]
** Viscount Headfort&lt;ref name=&quot;:1&quot; /&gt;
*** 4th: Thomas Taylour (6 December 1870 – 22 July 1894)
*** 5th: Geoffrey Thomas Taylour (22 July 1894 – 29 January 1943)
*Papers

==== Marquess of Sligo ====

* Did not attend the ball, but many people with the surname Browne attended a number of social events at about this time.
* Subsidiary Titles
** Earl of Altamont. Did not attend the ball; did not attend any social events analyzed so far.
** Earl of Clanricarde — Did not attend the ball but did attend a few social events about this time.
** Viscount of Westport&lt;ref name=&quot;:1&quot;&gt;&quot;Index to Viscounts and Viscountesses.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. 

https://www.thepeerage.com/index_viscount.htm.&lt;/ref&gt;
*** 5th: George John Browne (26 January 1845 – 30 December 1896), 5th Marquess
*** 6th: John Thomas Browne (30 December 1896 – 30 December 1903), 6th Marquess

==== Marquess of Waterford ====
* John Henry de La Poer Beresford, 5th Marquess of Waterford (1844–1895)
* Henry de La Poer Beresford, 6th Marquess of Waterford (1875–1911)&lt;ref&gt;{{Cite journal|date=2026-02-10|title=Henry Beresford, 6th Marquess of Waterford|url=https://en.wikipedia.org/w/index.php?title=Henry_Beresford,_6th_Marquess_of_Waterford&amp;oldid=1337565707|journal=Wikipedia|language=en}}&lt;/ref&gt;

* Did not attend the ball, but members of the Beresford family were prominent socially at about this time.
* Subsidiary Titles
** Viscount Tyrone

=== Earls and Countesses ===

==== Earl of Annesley ====

* Did not attend the ball but did attend a number of social events in the 1890s.
* Subsidiary Title
** Viscount Glerawly&lt;ref name=&quot;:1&quot; /&gt;: 6th: Hugh Annesley (10 August 1874 – 15 December 1908), 6th Earl of Annesley

==== Earl of Bessborough ====
* Frederick George Brabazon Ponsonby, 6th Earl of Bessborough (1815–1895)
* Walter William Brabazon Ponsonby, 7th Earl of Bessborough (1821–1906), would have been Viscount Duncannon 1880–1895
* Edward Ponsonby, 8th Earl of Bessborough (1851–1920), would have been Viscount Duncannon 1895–1906

* Did not attend the ball, but the [[Social Victorians/People/Ponsonby|Ponsonby]] family attended many social events at about this time, including mention of Lady Duncannon's school that taught fabric arts.
* Subsidiary Titles
** Viscount Duncannon

==== Earl of Caledon ====

* Did not attend the ball but did attend a number of social events about this time.
* James Alexander, 4th Earl of Caledon (1846–1898)&lt;ref&gt;{{Cite journal|date=2025-11-21|title=James Alexander, 4th Earl of Caledon|url=https://en.wikipedia.org/w/index.php?title=James_Alexander,_4th_Earl_of_Caledon&amp;oldid=1323312651|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Eric James Desmond Alexander, 5th Earl of Caledon (1885–1968), succeeded as earl in 1898.&lt;ref&gt;{{Cite journal|date=2025-11-21|title=Eric Alexander, 5th Earl of Caledon|url=https://en.wikipedia.org/w/index.php?title=Eric_Alexander,_5th_Earl_of_Caledon&amp;oldid=1323313583|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Subsidiary Title
** Viscount Caledon

==== Earl of Carrick ====

* Did not attend the ball.

==== Earl Castle Stewart ====

* Did not attend the ball.
* 5th Earl: Henry James Stuart-Richardson (12 September 1874 – 5 June 1914)&lt;ref&gt;&quot;Henry James Stuart-Richardson, 5th Earl Castle Stewart.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 2412 https://www.thepeerage.com/p12413.htm#i124125.&lt;/ref&gt;
* Subsidiary Title
** Viscount Castle Stewart

==== Earl of Cavan ====

* Did not attend the ball.

==== Earl of Clancarty ====

* Did not attend the ball and attended few social events researched so far.
* Richard Somerset Le Poer Trench, 4th Earl of Clancarty (1834–1891)&lt;ref&gt;{{Cite journal|date=2026-01-10|title=Richard Trench, 4th Earl of Clancarty|url=https://en.wikipedia.org/w/index.php?title=Richard_Trench,_4th_Earl_of_Clancarty&amp;oldid=1332219771|journal=Wikipedia|language=en}}&lt;/ref&gt;
* William Frederick Le Poer Trench, 5th Earl of Clancarty (1868–1929)&lt;ref&gt;{{Cite journal|date=2025-11-05|title=William Trench, 5th Earl of Clancarty|url=https://en.wikipedia.org/w/index.php?title=William_Trench,_5th_Earl_of_Clancarty&amp;oldid=1320532351|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Subsidiary Title
** Viscount Dunlo

==== [[Social Victorians/People/Clanwilliam|Earl and Countess of Clanwilliam]] ====

* Did not attend the ball.
* Subsidiary Title
** Viscount Clanwilliam&lt;ref name=&quot;:1&quot; /&gt;: 4th: Richard James Meade (7 October 1879 – 4 August 1907), 4th Earl

==== Earl of Cork, Earl of Orrery ====

* Cork and Orrery, did attend the ball.

==== Earl of Courtown ====

* Did not attend the ball.

==== Earl of Darnley ====
* John Bligh, 6th Earl of Darnley (1827–1896), British&lt;ref&gt;{{Cite journal|date=2026-02-07|title=John Bligh, 6th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=John_Bligh,_6th_Earl_of_Darnley&amp;oldid=1337113925|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Edward Bligh, 7th Earl of Darnley (1851–1900), Lord Clifton much of his adult life, &quot;English&quot;&lt;ref&gt;{{Cite journal|date=2026-05-05|title=Edward Bligh, 7th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=Edward_Bligh,_7th_Earl_of_Darnley&amp;oldid=1352607379|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Did not attend the ball, but the Bligh family attended some social events from about this time.
* Subsidiary Titles:
** Viscount Darnley

==== Earl of Desmond ====

* Did not attend the ball.

==== [[Social Victorians/People/Donoughmore|Earl of Donoughmore]] ====

* Did not attend the ball but did attend a number of social events about this time.
* John Luke George Hely-Hutchinson, 5th Earl of Donoughmore (1848–1900)&lt;ref&gt;{{Cite journal|date=2025-05-01|title=John Hely-Hutchinson, 5th Earl of Donoughmore|url=https://en.wikipedia.org/w/index.php?title=John_Hely-Hutchinson,_5th_Earl_of_Donoughmore&amp;oldid=1288332715|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Subsidiary Title
** Viscount Donoughmore

==== Earl of Drogheda ====

* Did not attend the ball.

* Subsidiary Titles
** Viscount Moore — no evidence of the Viscount or Viscountess Moore at social events at about this time.

==== Earl of Granard ====
* Did not attend the ball.
* Bernard Arthur William Patrick Hastings Forbes, 8th Earl of Granard (17 September 1874 – 10 September 1948)[https://en.wikipedia.org/wiki/Bernard_Forbes,_8th_Earl_of_Granard]
* Anglo-Irish
* Subsidiary Titles
** Bernard Arthur William Patrick Hastings Forbes, styled Viscount Forbes from 1874 to 1889

==== Earl of Kingston ====
* Did not attend the ball.
* Subsidiary Title
** Viscount Kingsborough (of Viscount Kingston of Kingborough, co. Sligo)&lt;ref name=&quot;:1&quot; /&gt;
*** 8th: Henry Newcomen King-Tenison (21 June 1871 – 13 January 1896)
*** 9th: Henry Edwyn King-Tenison (13 January 1896 – 11 January 1946)
**Viscount Lorton

==== Earl of Lisburne ====
* Did not attend the ball.
* Ernest Augustus Malet Vaughan, 5th Earl of Lisburne (1836–1888)&lt;ref&gt;{{Cite journal|date=2025-12-03|title=Ernest Augustus Malet Vaughan, 5th Earl of Lisburne|url=https://en.wikipedia.org/w/index.php?title=Ernest_Augustus_Malet_Vaughan,_5th_Earl_of_Lisburne&amp;oldid=1325511612|journal=Wikipedia|language=en}}&lt;/ref&gt;
** Owned a lot of land in Cardiganshire, Wales
** Conservative, but withdrew from politics
* George Henry Arthur Vaughan, 6th Earl of Lisburne (1862–1899)
* Ernest Edmund Henry Malet Vaughan, 7th Earl of Lisburne (1892–1965)
** Welsh nobleman, of Trawsgoed, Cardiganshire. 7 years old when he succeeded to the earldom

==== Earl of Longford ====

* Did not attend the ball.

==== Earl and Countess of Meath ====

* Did not attend the ball.

==== Earl of Mexborough ====

* Did not attend the ball

==== Earl of Mornington ====

* Subsidiary title of the Duke of Wellington (in the peerage of the UK).

==== Earl of Normanton ====

* Did not attend the ball, but did attend some social events in the 1880s and 1890s.
* James Charles Herbert Welbore Ellis Agar, 3rd Earl of Normanton (1818–1896)&lt;ref&gt;{{Cite journal|date=2025-10-06|title=James Agar, 3rd Earl of Normanton|url=https://en.wikipedia.org/w/index.php?title=James_Agar,_3rd_Earl_of_Normanton&amp;oldid=1315461436|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Sidney James Agar, 4th Earl of Normanton (1865–1933)&lt;ref&gt;{{Cite journal|date=2026-05-19|title=Sidney James Agar, 4th Earl of Normanton|url=https://en.wikipedia.org/w/index.php?title=Sidney_James_Agar,_4th_Earl_of_Normanton&amp;oldid=1355064165|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Subsidiary Title
** Viscount Somerton

==== Earl of Portarlington ====

* Did not attend the ball. Members of this family attended a few social events at about this time.
* Subsidiary Title
** Viscount Carlow&lt;ref name=&quot;:1&quot; /&gt;
*** 5th: Lionel Seymour William Dawson-Damer (1 March 1889 – 17 December 1892), Earl of Portarlington
*** 6th: Lionel George Henry Seymour Dawson-Damer (17 December 1892 – 31 August 1900)

==== Earl of Roden ====

* Did not attend the ball.
* Subsidiary Title
** Viscount Jocelyn&lt;ref name=&quot;:1&quot; /&gt;
*** 6th: John Strange Jocelyn (9 January 1880 – 3 July 1897)
*** 7th: William Henry Jocelyn (3 July 1897 – 23 January 1910)

==== Earl of Shannon ====

* Did not attend the ball.

==== Earl of Shelburne ====

* Subsidiary title of the Marquess of Lansdowne (in the peerage of Great Britain).
* Did not attend the ball, and did not attend any social events analyzed so far.

==== Earl of Tyrone ====

* Did not attend

==== Earl of Waterford ====

* Not a subsidiary title of the Marquess of Waterford but of the Earl of Shrewsbury in the peerage of England.

==== Earl of Westmeath ====

* Did not attend the ball.

==== Earl of Winterton ====

* Did not attend the ball.

=== Viscounts and Viscountesses ===

==== Viscount Ashbrook ====
* William Spencer Flower, 7th Viscount Ashbrook (1830–1906)&lt;ref&gt;{{Cite journal|date=2025-12-01|title=Viscount Ashbrook|url=https://en.wikipedia.org/w/index.php?title=Viscount_Ashbrook&amp;oldid=1325071512|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Did not attend the ball, has no social presence at about this time.

==== Viscount Banger ====

* Did not attend the ball but attended a few social events at about this time.
* Edward Ward, 4th Viscount Bangor (1827–1881)&lt;ref&gt;{{Cite journal|date=2026-03-16|title=Edward Ward, 4th Viscount Bangor|url=https://en.wikipedia.org/w/index.php?title=Edward_Ward,_4th_Viscount_Bangor&amp;oldid=1343882576|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Henry William Crosbie Ward, 5th Viscount Bangor (1828–1911)&lt;ref&gt;{{Cite journal|date=2026-03-02|title=Henry Ward, 5th Viscount Bangor|url=https://en.wikipedia.org/w/index.php?title=Henry_Ward,_5th_Viscount_Bangor&amp;oldid=1341354058|journal=Wikipedia|language=en}}&lt;/ref&gt;

==== Viscount Boyne ====

* Did not attend the ball, but did attend a number of events at about this time.

==== Viscount Callan ====

* Did not attend the ball, and does not have much if any social presence at about this time.
* The Viscount Callan is a subsidiary title of the Earl of Denbigh in the Peerage of England.

==== Viscount Charlemont ====
* Did not attend the ball.
* Colonel James Alfred Caulfeild, 7th Viscount Charlemont (20 March 1830 – 4 July 1913), Irish&lt;ref&gt;{{Cite journal|date=2026-05-02|title=James Caulfeild, 7th Viscount Charlemont|url=https://en.wikipedia.org/w/index.php?title=James_Caulfeild,_7th_Viscount_Charlemont&amp;oldid=1352129469|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Unionist

==== Viscount Chetwynd ====
* Does not seem to have attended the ball, but Chetwynds were socially very active at about this time.
* Godfrey John Boyle Chetwynd, 8th Viscount Chetwynd (1863–1936), British&lt;ref&gt;{{Cite journal|date=2026-05-24|title=Godfrey Chetwynd, 8th Viscount Chetwynd|url=https://en.wikipedia.org/w/index.php?title=Godfrey_Chetwynd,_8th_Viscount_Chetwynd&amp;oldid=1355878192|journal=Wikipedia|language=en}}&lt;/ref&gt;

==== Viscount de Vesci ====

* Did not attend the ball but attended several social events at about this time.
* 4th Viscount de Vesci: John Robert William Vesey (23 December 1875 – 6 July 1903)&lt;ref name=&quot;:1&quot; /&gt;
* &quot;The family seat was Abbeyleix House, near Abbeyleix, County Laois.&quot;&lt;ref&gt;{{Cite journal|date=2026-02-09|title=Viscount de Vesci|url=https://en.wikipedia.org/w/index.php?title=Viscount_de_Vesci&amp;oldid=1337491855|journal=Wikipedia|language=en}}&lt;/ref&gt;

==== Viscount Dillon ====

* Did not attend the ball, but several Dillons attended other social events at about this time.

==== Viscount Doneraile&lt;ref&gt;{{Cite journal|date=2026-01-16|title=Viscount Doneraile|url=https://en.wikipedia.org/w/index.php?title=Viscount_Doneraile&amp;oldid=1333262628|journal=Wikipedia|language=en}}&lt;/ref&gt; ====

* Did not attend the ball, but did attend the Warwick Bal Poudré and few other social events at about this time.
* Hayes St Leger, 4th Viscount Doneraile (1818–1887)
* Richard Arthur St Leger, 5th Viscount Doneraile (1825–1891)
* Edward St Leger, 6th Viscount Doneraile (1866–1941)

==== [[Social Victorians/People/Downe|Viscount Downe]] ====
* Did not attend the ball but attended many social events at about this time.
* Major-General Hugh Richard Dawnay, 8th Viscount Downe (20 July 1844 – 21 January 1924)&lt;ref&gt;{{Cite journal|date=2026-03-24|title=Hugh Dawnay, 8th Viscount Downe|url=https://en.wikipedia.org/w/index.php?title=Hugh_Dawnay,_8th_Viscount_Downe&amp;oldid=1345146095|journal=Wikipedia|language=en}}&lt;/ref&gt;
* British Army general

==== Viscount Ferrard ====

* See Viscount Massereene, below. By the end of the century, it was the Viscount and Viscountess of Massereene and Ferrard.

==== Viscount Fitzmaurice ====

* A subsidiary title of the Marquess of Lansdowne (in the Peerage of Great Britain).
* 6th Viscount Fitzmaurice, Henry Charles Keith Petty-FitzMaurice (5 July 1866 – 3 June 1927)&lt;ref&gt;&quot;Henry Charles Keith Petty-FitzMaurice, 5th Marquess of Lansdowne.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 958

https://www.thepeerage.com/p959.htm#i9586.&lt;/ref&gt;

==== Viscount Gage ====
* Henry Charles Gage, 5th Viscount Gage (1854–1912)&lt;ref&gt;{{Cite journal|date=2025-06-21|title=Viscount Gage|url=https://en.wikipedia.org/w/index.php?title=Viscount_Gage&amp;oldid=1296646030|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Did not attend the ball, but members of this family attended a number of social events at about this time.

==== Viscount Galway ====
* George Edmund Milnes Monckton-Arundell, 7th Viscount Galway (1844–1931), British conservative&lt;ref&gt;{{Cite journal|date=2025-08-08|title=George Monckton-Arundell, 7th Viscount Galway|url=https://en.wikipedia.org/w/index.php?title=George_Monckton-Arundell,_7th_Viscount_Galway&amp;oldid=1304770631|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Did not attend the ball, but Viscount and Viscountess Galway attended many social events at about this time.
* Subsidiary Title
** Baron Monckton (in the Peerage of the United Kingdom)

==== Viscount Gormanston ====

* Did not attend the ball, has no social presence in the late 19th-century newspapers at this time.

==== [[Social Victorians/People/Gort|Viscount Gort]] ====

* Did not attend the ball, but attended some social events at about this time.
* Standish Prendergast Vereker, 4th Viscount Gort (1819–1900)&lt;ref&gt;&quot;Standish Prendergast Vereker, 4th Viscount Gort.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 4626 https://www.thepeerage.com/p4627.htm#i46261.&lt;/ref&gt;
* John Gage Prendergast Vereker, 5th Viscount Gort (1849–1902)&lt;ref&gt;{{Cite journal|date=2025-05-28|title=John Vereker, 5th Viscount Gort|url=https://en.wikipedia.org/w/index.php?title=John_Vereker,_5th_Viscount_Gort&amp;oldid=1292670203|journal=Wikipedia|language=en}}&lt;/ref&gt;

==== Viscount Grandison ====

* Did not attend the ball, has no social presence in the late 19th-century newspapers at this time.
* The Viscount Grandison is a subsidiary title of the Earl of Jersey in the Peerage of England.

==== Viscount Grimston ====
* Subsidiary title of the Earl of Verulam (in the Peerage of the United Kingdom)
* Did not attend the ball, but a number of members of this family attended social events at about this time.

==== Viscount Harberton ====

* Did not attend the ball; Viscountess Harberton is mentioned once in social events at about this time so far.
* James Spencer Pomeroy, 6th Viscount Harberton (1836–1912)&lt;ref&gt;&quot;James Spencer Pomeroy, 6th Viscount Harberton.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person Page 4315 https://www.thepeerage.com/p43151.htm#i431502.&lt;/ref&gt;
* Florence Wallace Pomeroy, Viscountess Harberton (1843–1911), suffragette, cyclist, President of the Rational Dress Society&lt;ref&gt;{{Cite journal|date=2026-03-12|title=Florence Wallace Pomeroy, Viscountess Harberton|url=https://en.wikipedia.org/w/index.php?title=Florence_Wallace_Pomeroy,_Viscountess_Harberton&amp;oldid=1343082631|journal=Wikipedia|language=en}}&lt;/ref&gt;

==== Viscount Lifford ====

* Did not attend the ball; the only social event at about this time so far is the Queen's Diamond Jubilee garden party.
* James Hewitt, 4th Viscount Lifford (1811–1887)&lt;ref&gt;{{Cite journal|date=2025-09-11|title=James Hewitt, 4th Viscount Lifford|url=https://en.wikipedia.org/w/index.php?title=James_Hewitt,_4th_Viscount_Lifford&amp;oldid=1310741456|journal=Wikipedia|language=en}}&lt;/ref&gt;
* James Wilfrid Hewitt, 5th Viscount Lifford (12 October 1837 – 20 March 1913)&lt;ref&gt;&quot;James Wilfrid Hewitt, 5th Viscount Lifford.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person Page 2227 https://www.thepeerage.com/p22271.htm#i222701.&lt;/ref&gt;

==== Earl of Listowel ====
* Pronounced &quot;Lish-''toe''-ell.&quot;&lt;ref&gt;{{Cite journal|date=2024-10-15|title=Earl of Listowel|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Listowel&amp;oldid=1251322273|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Did not attend the ball, but hosted and attended social events at about this time.
* William Hare, 3rd Earl of Listowel (1833–1924)&lt;ref&gt;{{Cite journal|date=2026-04-17|title=William Hare, 3rd Earl of Listowel|url=https://en.wikipedia.org/w/index.php?title=William_Hare,_3rd_Earl_of_Listowel&amp;oldid=1349570352|journal=Wikipedia|language=en}}&lt;/ref&gt;, Irish peer
* Subsidiary Title
** Viscount Ennismore and Listowel
** Baron Ennismore

==== Viscount Massereene ====
* Did not attend the ball but did attend a few events at about this time. See Viscount Ferrard, above. By the end of the century, it was the Viscount and Viscountess of Massereene and Ferrard.
* Anglo-Irish
* Clotworthy John Eyre Skeffington, 11th Viscount Massereene (9 October 1842 – 26 June 1905)&lt;ref&gt;{{Cite journal|date=2024-11-23|title=Clotworthy Skeffington, 11th Viscount Massereene|url=https://en.wikipedia.org/w/index.php?title=Clotworthy_Skeffington,_11th_Viscount_Massereene&amp;oldid=1259199982|journal=Wikipedia|language=en}}&lt;/ref&gt; and 4th Viscount Ferrard (28 April 1863 – 26 June 1905)

==== Viscount Molesworth ====
* Did not attend the ball, but attended the Warwick Bal Poudré and a number of other social events at about this time.
* Samuel Molesworth, 8th Viscount Molesworth (1829–1906), may have been a Quaker

==== Viscount Monck ====

* Did not attend the ball, but attended a number of social events at about this time.
* Charles Stanley Monck, 4th Viscount Monck (1819–1894)&lt;ref&gt;{{Cite journal|date=2026-04-05|title=Charles Monck, 4th Viscount Monck|url=https://en.wikipedia.org/w/index.php?title=Charles_Monck,_4th_Viscount_Monck&amp;oldid=1347311992|journal=Wikipedia|language=en}}&lt;/ref&gt;, British
* Henry Power Charles Stanley Monck, 5th Viscount Monck (1849–1927)&lt;ref&gt;&quot;Henry Power Charles Stanley Monck, 5th Viscount Monck of Ballytrammon.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 3880 https://www.thepeerage.com/p3881.htm#i38802.&lt;/ref&gt;

==== Viscount Mountgarret ====

* Did not attend the ball, has no social presence in the late 19th-century newspapers at this time.

==== [[Social Victorians/People/Powerscourt|Viscount Powerscourt]] ====
* Mervyn Wingfield, 7th Viscount Powerscourt (1836–1904)&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2026-02-18|title=Mervyn Wingfield, 7th Viscount Powerscourt|url=https://en.wikipedia.org/w/index.php?title=Mervyn_Wingfield,_7th_Viscount_Powerscourt&amp;oldid=1339057453|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Did not attend the ball, but members of this family attended a number of social events at about this time.
* Subsidiary Title
** Baron Powerscourt (in the Peerage of the United Kingdom), 1885&lt;ref name=&quot;:0&quot; /&gt;

==== Viscount Southwell ====

* Did not attend the ball, though the Viscount and Viscountess attended a few social events at about this time.
* 5th&lt;ref name=&quot;:1&quot; /&gt;: Arthur Robert Pyers Southwell (26 April 1878 – 5 October 1944)&lt;ref&gt;&quot;Arthur Robert Pyers Southwell, 5th Viscount Southwell of Castle Mattress.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page

https://www.thepeerage.com/p7550.htm#i75497.&lt;/ref&gt;

==== Viscount Valentia ====

* Did not attend the ball, attended some social events at about this time. Was on the Welcome Council for the 1887 American Exhibition.

=== Barons and Baronesses ===
Not all the barons extant at the end of the 19th century and listed on the Wikipedia [[wikipedia:Peerage_of_Ireland|Peerage of Ireland]] page are here — only the ones who were active socially.

==== Baron Conway and Killultagh ====

* Did not attend the ball, but people from the Conway and Seymour families attended a number of social events at about this time.
* Subsidiary title of the Marquess of Hertford (in the Peerage of England and Great Britain).
* Francis Hugh George Seymour, 5th Marquess of Hertford (1812–1884)&lt;ref&gt;{{Cite journal|date=2026-04-05|title=Francis Seymour, 5th Marquess of Hertford|url=https://en.wikipedia.org/w/index.php?title=Francis_Seymour,_5th_Marquess_of_Hertford&amp;oldid=1347294689|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Hugh de Grey Seymour, 6th Marquess of Hertford (1843–1912)&lt;ref&gt;{{Cite journal|date=2026-04-05|title=Hugh Seymour, 6th Marquess of Hertford|url=https://en.wikipedia.org/w/index.php?title=Hugh_Seymour,_6th_Marquess_of_Hertford&amp;oldid=1347303090|journal=Wikipedia|language=en}}&lt;/ref&gt;

==== Baron Digby ====

* Did not attend the ball, but people from this family attended a number of social events at about this time.
* Edward St Vincent Digby, 9th and 3rd Baron Digby (1809–1889)&lt;ref&gt;{{Cite journal|date=2025-12-15|title=Edward Digby, 9th Baron Digby|url=https://en.wikipedia.org/w/index.php?title=Edward_Digby,_9th_Baron_Digby&amp;oldid=1327712265|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Edward Henry Trafalgar Digby, 10th and 4th Baron Digby (1846–1920)&lt;ref&gt;{{Cite journal|date=2026-01-26|title=Edward Digby, 10th Baron Digby|url=https://en.wikipedia.org/w/index.php?title=Edward_Digby,_10th_Baron_Digby&amp;oldid=1334892957|journal=Wikipedia|language=en}}&lt;/ref&gt;

==== Baron Inchiquin ====

* Did not attend the ball, but people from this family attended a number of social events at about this time.
* Edward Donough O'Brien, 14th Baron Inchiquin (1839–1900)&lt;ref&gt;{{Cite journal|date=2026-04-28|title=Edward O'Brien, 14th Baron Inchiquin|url=https://en.wikipedia.org/w/index.php?title=Edward_O%27Brien,_14th_Baron_Inchiquin&amp;oldid=1351543832|journal=Wikipedia|language=en}}&lt;/ref&gt;

== Peerage of the United Kingdom of Great Britain and Ireland ==
After the forced 1801 Act of Union.

=== Earls and Countesses ===

==== Earl of Limerick ====

* Did not attend the ball, but did attend a number of events at about this time.

==== Earl of Norbury ====

* Did not attend the ball, but attended some social events at about this time.
* Subsidiary Title
** Baron Norbury

==== Earl of Ranfurly ====

* Did not attend the ball, and they have a small social presence in the newspapers in the 1880s and 1890s.

==== Earl of Rosse ====

* Did not attend the ball, but did attend a few events at about this time.

== Irish Nationalists ==

== Irish Unionists ==

== Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball ==

==== [[Social Victorians/People/Abercorn|Duke and Duchess of Abercorn]] ====
* This dukedom is in the peerage of the United Kingdom of Great Britain and Ireland

* James Hamilton, 1st Duke of Abercorn (1811–1885), elder son of Lord Hamilton, &quot;styled Viscount Hamilton from 1814 to 1818 and The Marquess of Abercorn from 1818 to 1868, was a Conservative statesman who twice served as Lord Lieutenant of Ireland.&quot;&lt;ref&gt;{{Cite journal|date=2026-04-05|title=James Hamilton, 1st Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_1st_Duke_of_Abercorn&amp;oldid=1347253763|journal=Wikipedia|language=en}}&lt;/ref&gt;
* James Hamilton, 2nd Duke of Abercorn (1838–1913), eldest son of the 1st Duke, &quot;styled Viscount Hamilton until 1868 and Marquess of Hamilton from 1868 to 1885, was a British nobleman, courtier, and diplomat.&quot;&lt;ref&gt;{{Cite journal|date=2026-01-25|title=James Hamilton, 2nd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_2nd_Duke_of_Abercorn&amp;oldid=1334676058|journal=Wikipedia|language=en}}&lt;/ref&gt;

* Subsidiary Titles
** Marquess of Abercorn
** Viscount Hamilton
** Viscount Strabane, county Tyrone
*Papers: PRONI for the Abercorn papers [GB 0255 PRONI/D623]; some individuals' papers (the Tighe Hamilton Howard papers, https://iar.ie/archive/tighe-hamilton-howard-papers) from the Hamilton family are in the National Library of Ireland.

==== [[Social Victorians/People/Londonderry|Marquess and Marchioness of Londonderry]] ====
* The Marquess and Marchioness attended the ball, she led one of the courts as Maria Thérèse, plus two of their children attended, one of whom is Viscount Castlereagh.
* Charles Stewart Vane-Tempest-Stewart, 6th Marquess of Londonderry&lt;ref&gt;&quot;Charles Stewart Vane-Tempest-Stewart, 6th Marquess of Londonderry.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 1277 https://www.thepeerage.com/p1278.htm#i12772.&lt;/ref&gt;
* Lady Theresa Susey Helen Chetwynd-Talbot, Marchioness of Londonderry&lt;ref&gt;&quot;Lady Theresa Susey Helen Chetwynd-Talbot.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 1277 https://www.thepeerage.com/p1278.htm#i12771.&lt;/ref&gt;
* Subsidiary Titles
** [[Social Victorians/People/Londonderry|Earl of Londonderry]]
** Viscount Castlereagh — Charles Stewart Henry Vane-Tempest-Stewart (6 November 1884 – 8 February 1915)
*Papers
**In PRONI [GB 0255 PRONI/D2846]: &quot;The Theresa, Lady Londonderry Papers comprise c.4,600 papers and 15 volumes of diaries, scrapbooks, etc, 1858-1919, mainly of Theresa, Marchioness of Londonderry (1856-1919), wife/widow of the 6th Marquess, but including some papers of the 6th Marquess himself, of and about his mother, Mary Cornelia, widow of the 5th Marquess, and of his brothers Lords Henry and Herbert Vane-Tempest.&quot;&lt;ref&gt;{{Cite web|url=https://iar.ie/archive/theresa-lady-londonderry-papers/|title=Theresa, Lady Londonderry Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}&lt;/ref&gt;
**In PRONI [GB 0255 PRONI/D3099]: the &quot;Papers of the 7th Marquess of Londonderry and his wife Edith&quot; collection also hold the papers of Edith's father, [[Social Victorians/People/Henry Chaplin|Henry, 1st Viscount Chaplin]], who attended the ball, as did she and a brother. [D3099/1 Henry, 1st Viscount Chaplin, father-in-law of 7th Marquess of Londonderry. Political and personal papers; D3099/3 Edith Helen Chaplin, wife of 7th Marquess of Londonderry. Personal letters and papers]&lt;ref&gt;{{Cite web|url=https://iar.ie/archive/papers-7th-marquess-londonderry-wife-edith/|title=Papers of the 7th Marquess of Londonderry and his wife Edith|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}&lt;/ref&gt;

==== [[Social Victorians/People/Lucan|Earl of Lucan]] ====

* Some members of the family attended the ball, and the family attended a number of social events at this time.
* Papers

==== [[Social Victorians/People/Ormonde|Marquess and Marchioness of Ormonde]] ====
* James Edward Butler, 3rd Marquess of Ormonde and 21st Earl of Ormonde (1844–1919)&lt;ref&gt;{{Cite journal|date=2026-05-03|title=Earl of Ormond (Ireland)|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Ormond_(Ireland)&amp;oldid=1352334266|journal=Wikipedia|language=en}}&lt;/ref&gt; Now extinct; earldom dormant. Castle X was their manor, but they don't appear to have any papers.

* Subsidiary Titles
* Papers

==== [[Social Victorians/People/Antrim|Earl of Antrim]] ====

* The earl and countess did not attend the ball, but two of his brothers did.
* Papers
** [https://iar.ie/archive/earl-antrim-estate-papers/ Estate papers of the Earls of Antrim] [GB 0255 PRONI/D2977] are in PRONI. I don't see personal papers listed, but the collection has 50,000 documents 1603–1967.
** Also &quot;D4091 Papers of Sir Schomberg MacDonnell, Louisa Countess of Antrim and the Stuart family of Dalness. MIC615 The diaries of Louisa, Countess of Antrim.&quot;&lt;ref&gt;{{Cite web|url=https://iar.ie/archive/earl-antrim-estate-papers/|title=Earl of Antrim Estate Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}&lt;/ref&gt;

==== [[Social Victorians/People/Arran|Earl of Arran]] ====

* Attended the ball.
* Subsidiary Titles
** Viscount Sudley: 5th: Arthur Saunders William Charles Fox Gore (25 Jun 1884-14 Mar 1901), 5th Earl of Arran&lt;ref name=&quot;:1&quot; /&gt;
*Papers

==== [[Social Victorians/People/Belmore|Earl Belmore]] ====

* Did not attend the ball, although [[Social Victorians/People/Rowton|Montagu Lowry-Corry, 1st Baron Rowton]] did, but did attend a number of social events about this time.
* 4th Earl: Somerset Richard Lowry-Corry (17 Dec 1845-6 Apr 1913)&lt;ref&gt;{{Cite journal|date=2026-04-17|title=Somerset Lowry-Corry, 4th Earl Belmore|url=https://en.wikipedia.org/w/index.php?title=Somerset_Lowry-Corry,_4th_Earl_Belmore&amp;oldid=1349375684|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Subsidiary Title
** Viscount Belmore (though the subsidiary title for the heir apparent is Viscount Corry?)
*Papers

==== [[Social Victorians/People/Dunraven|Earl of Dunraven and Mount-Earl]] ====

* The [[Social Victorians/People/Dunraven|Earl of Dunraven and Mount-Earl]] and Countess of Dunraven, and their daughter Lady Aileen May Wyndham-Quin attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House.
* Windham Wyndham-Quin, 4th Earl of Dunraven and Mount-Earl (1841–1926)&lt;ref&gt;{{Cite journal|date=2026-05-22|title=Windham Wyndham-Quin, 4th Earl of Dunraven and Mount-Earl|url=https://en.wikipedia.org/w/index.php?title=Windham_Wyndham-Quin,_4th_Earl_of_Dunraven_and_Mount-Earl&amp;oldid=1355461019|journal=Wikipedia|language=en}}&lt;/ref&gt;, Anglo-Irish
* Papers

==== [[Social Victorians/People/Cole|Earl and Countess of Enniskillen]] ====

* The Earl and Countess and a daughter attended the ball. Papers in PRONI.
* Subsidiary Title
** 4th Viscount Enniskillen: Lowry Egerton Cole (12 November 1886 – 28 April 1924)&lt;ref name=&quot;:1&quot; /&gt;
*Papers

==== [[Social Victorians/People/Crichton|Earl of Erne]] ====

* Some members of the family attended the ball. Papers in PRONI.
* The newspapers were very inconsistent in the spelling of the family name Crichton.
* Subsidiary Title
** Viscount Erne&lt;ref name=&quot;:1&quot; /&gt;
*** 3rd Earl of Erne: John Crichton (10 June 1842 – 3 October 1885)
*** 4th Earl of Erne: John Henry Crichton (3 October 1885 – 2 December 1914)
*Papers

==== [[Social Victorians/People/Gosford|Earl of Gosford]] ====

* The Earl and Countess of Gosford attended the ball, as did a son and a daughter. They attended many social events at about this time.
* Subsidiary Title
** Viscount Gosford of Market Hill, co. Armagh&lt;ref name=&quot;:1&quot; /&gt;
*** 5th Earl of Gosford: Archibald Brabazon Sparrow Acheson (15 June 1864 – 11 April 1922)
*Papers

==== Earl of Kerry ====
* Subsidiary title of the [[Social Victorians/People/Lansdowne|Marquess of Lansdowne]] (in the peerage of Great Britain). Attended the ball.
* Subsidiary Titles
** Viscount Clanmaurice
*Papers

==== [[Social Victorians/People/Kilmorey|Earl of Kilmorey]] ====

* Anglo-Irish
* Nellie Countess of Kilmorey attended the ball; Francis, 3rd Earl was alive at the time, did he attend? Both he and she attended a number of social events from about this time.
* Papers

==== [[Social Victorians/People/Mayo|Earl of Mayo]] ====

* Some members of the family attended the ball.
* Viscount Mayo of Monycrower, co. Mayo&lt;ref name=&quot;:1&quot; /&gt;
** 7th Earl of Mayo: Dermot Robert Wyndham Bourke (8 February 1872 – 31 December 1927)
*Papers

==== [[Social Victorians/People/Midleton|Viscount Midleton]] ====
* Some people from this family seem to have attended the ball as well as many other social events at about this time.
* William Brodrick, 8th Viscount Midleton (6 January 1830 – 18 April 1907), &quot;Irish peer, landowner and Conservative politician in both Houses of Parliament&quot;&lt;ref&gt;{{Cite journal|date=2025-01-05|title=William Brodrick, 8th Viscount Midleton|url=https://en.wikipedia.org/w/index.php?title=William_Brodrick,_8th_Viscount_Midleton&amp;oldid=1267418489|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Sight and hearing disabilities caused by intermarriage. A daughter became a Republican.
* Papers

==== Baron Carrington ====
* [[Social Victorians/People/Carrington|Charles Robert Wynn-Carington, 1st Marquess of Lincolnshire]] (1843–1928) attended the ball.
* Baron Carrington is a subsidiary title of the Marquess of Lincolnshire (created in 1912; Earl Carrington created in 1895).&lt;ref&gt;{{Cite journal|date=2026-05-20|title=Baron Carrington|url=https://en.wikipedia.org/w/index.php?title=Baron_Carrington&amp;oldid=1355207880|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Papers

==== Baron Dufferin and Claneboye&lt;ref&gt;{{Cite journal|date=2026-02-07|title=Baron Dufferin and Claneboye|url=https://en.wikipedia.org/w/index.php?title=Baron_Dufferin_and_Claneboye&amp;oldid=1337113957|journal=Wikipedia|language=en}}&lt;/ref&gt; ====

* Members of this family did attend the ball as well as many social events at about this time.
* [[Social Victorians/People/Hamilton Temple Blackwood|Frederick Temple Hamilton-Temple-Blackwood]], 1st Marquess of Dufferin and Ava (1826–1902)&lt;ref&gt;{{Cite journal|date=2026-05-27|title=Frederick Hamilton-Temple-Blackwood, 1st Marquess of Dufferin and Ava|url=https://en.wikipedia.org/w/index.php?title=Frederick_Hamilton-Temple-Blackwood,_1st_Marquess_of_Dufferin_and_Ava&amp;oldid=1356387854|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Papers

==== Baron Garvagh ====

* [[Social Victorians/People/Garvagh|Florence Canning, Lady Garvagh]] attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House.
* Charles John Spencer George Canning, 3rd Baron Garvagh (1852–1915)&lt;ref&gt;{{Cite journal|date=2026-02-06|title=Baron Garvagh|url=https://en.wikipedia.org/w/index.php?title=Baron_Garvagh&amp;oldid=1336941309|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Papers

==== Baron Rossmore of Monaghan ====

* A [[Social Victorians/People/Naylor|Miss Naylor]] (Lady Rossmore's sister) of this family attended the ball.
* Derrick Warner William Westenra, 5th Baron Rossmore (1853–1921)&lt;ref&gt;{{Cite journal|date=2024-08-27|title=Derrick Westenra, 5th Baron Rossmore|url=https://en.wikipedia.org/w/index.php?title=Derrick_Westenra,_5th_Baron_Rossmore&amp;oldid=1242602083|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Papers

== References ==
{{reflist}}</text>
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      <text bytes="445" sha1="3ur6l6f76dmxrh4b7jjuckwuk3cnth2" xml:space="preserve">{{Merge|Igbo culture}}{{welcome and expand}}
== Introduction ==
The married and unmarried daughters of a particular clan or village in [[w:ala Igbo|ala Igbo]].

While Igbo society is about tracing descent through the male, Ndi Umuada serve as a vital checks and balances in the society. They represent a collective authority that balances the political power held by the males. Their words and decisions are highly respected and are often final.</text>
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  <page>
    <title>User talk:Wmbata</title>
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        <username>Atcovi</username>
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      <text bytes="3493" sha1="fg1w2qddkgkgz5yz5hrsugwznvbjxl4" xml:space="preserve">==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]],  Wmbata!'''|width=100%}}
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See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:56, 5 June 2026 (UTC)&lt;/div&gt;
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:Thanks @[[User:Jtneill|Jtneill]] [[User:Wmbata|Wmbata]] ([[User talk:Wmbata|discuss]] • [[Special:Contributions/Wmbata|contribs]]) 13:22, 5 June 2026 (UTC)

== [[Igbo culture]] ==

Hi Wmbata, and welcome to our project! Thanks for your contributions so far.

I wanted to give you some advice on how you can tailor your new contributions so it fits Wikiversity's [[Wikiversity:Scope|scope]]. For [[Igbo culture]], you could frame this page like a course so it could better fit the idea of a [[Wikiversity:Learning projects|learning project]]. Here's some examples of courses on Wikiversity that you can model:
*[[Algebra 1]] - introduction, course outline, quizzes listed for each course.
*[[Human Legacy Course]] - introduction, course outline, subpages listed, quizzes are there for certain pages.

Here are some other useful links that may be good for you to read:
*[[Wikiversity:Introduction]]
*[[Wikiversity:What is Wikiversity?]]
*[[Wikiversity:Differences between Wikiversity and Wikipedia]]

Please let me know if you have any questions, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:10, 6 June 2026 (UTC)</text>
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      <text bytes="3695" sha1="qoejeoli3wlxf99rpcn33fl4g8v6ol4" xml:space="preserve">==Welcome==
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Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.

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* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
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:Thanks @[[User:Jtneill|Jtneill]] [[User:Wmbata|Wmbata]] ([[User talk:Wmbata|discuss]] • [[Special:Contributions/Wmbata|contribs]]) 13:22, 5 June 2026 (UTC)

== [[Igbo culture]] ==

Hi Wmbata, and welcome to our project! Thanks for your contributions so far.

I wanted to give you some advice on how you can tailor your new contributions so it fits Wikiversity's [[Wikiversity:Scope|scope]]. For [[Igbo culture]], you could frame this page like a course so it could better fit the idea of a [[Wikiversity:Learning projects|learning project]]. Here's some examples of courses on Wikiversity that you can model:
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Please let me know if you have any questions, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:10, 6 June 2026 (UTC)

:Thanks , I've fixed it. If I have any more questions, i will let you know  [[User:Wmbata|Wmbata]] ([[User talk:Wmbata|discuss]] • [[Special:Contributions/Wmbata|contribs]]) 06:22, 7 June 2026 (UTC)</text>
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      <text bytes="21085" sha1="411v64aoqgpcvd49hlo45snq8zrb3gi" xml:space="preserve">{{Short description|Cultural traditions of the Igbo people}}
'''Igbo culture''' are the customs, practices and traditions of the Igbo people of southeastern Nigeria. It consists of ancient practices known as ''Odinala'' ''ndi'' ''igbo'' as well as new concepts added into the Igbo culture either by cultural evolution or by outside influence. These customs and traditions includes the Igbo people's visual art, music and dance forms, as well as their attire, Food, cuisine and language dialects. Because of their various subgroups, the variety of their culture is heightened further.

== Music ==
[[File:Udu.jpg|thumb|right|95px|Udu, an Igbo instrument]]
The Igbo people have a melodic and symphonic musical style. Instruments include Ọ̀pì  otherwise known as '''Oja''' a wind instrument similar to the flute, '''igba''', and '''ichaka'''.

Another popular musical form among Igbo people is highlife, which is a fusion of jazz and traditional music and widely popular in West Africa. The modern Igbo highlife is seen in the works of Prince Nico Mbarga, Dr Sir Warrior, Oliver De Coque, Bright Chimezie, Celestine Ukwu,Chief Osita Osadebe, And many others who are some of the greatest Igbo highlife musicians of the twentieth century. There are also other notable Igbo highlife artists, like the Mike Ejeagha, Paulson Kalu, Ali Chukwuma, Ozoemena Nwa Nsugbe.

== Art ==
Igbo art is known for various types of masquerades, masks, outfits (symbolizing people), animals and abstract conceptions. Igbo art is also known for its bronze castings found in the town of Igbo Ukwu from the 9th century.

&lt;gallery widths=&quot;200&quot; heights=&quot;200&quot; mode=&quot;packed&quot;&gt;
File:Nigeria, igbo, maschera-elmo della società mmuo, xx secolo.jpg|Helmet-mask; 20th century; Indianapolis Museum of Art (USA)
File:Nigeria, igbo, figura femminile per un tempietto, xx secolo.jpg|Female figure for a small temple, 20th century; Indianapolis Museum of Art
File:Igbo brass anklet.jpg|Anklet beaten from a solid brass bar of the type worn by Igbo women. Now in the collection of Wolverhampton Art Gallery. The leg-tube extends approximately 7&amp;nbsp;cm each side of the 35&amp;nbsp;cm disc.
File:Bronze ceremonial vessel in form of a snail shell, 9th century, Igbo-Ukwu, Nigeria.JPG|Bronze ceremonial vessel in form of a snail shell; 9th century; from Igbo-Ukwu; Nigerian National Museum (Lagos, Nigeria)
File:Eze Onyiudo (2).jpg|Eze Onyiudo Masquerade Awka-Etiti
&lt;/gallery&gt;

== Mythology ==
While today many Igbo people are Christian, the traditional ancient Igbo religion is known as Odinani. In the Igbo mythology, which is part of their ancient religion, the supreme God is called Chineke (&quot;the God of creation&quot;); Chineke created the world and everything in it and is associated with all things on Earth. To the ancient Igbo, the cosmos is divided into four complex parts:
 
* OKIKE (Creation)
* ALUSI (Supernatural Forces or Deities)
* MMUO (Spirit)
* UWA (World)

=== Alusi ===
[[File:Complex sculpture Nigeria BM Af1954 23 522 img02.jpg|thumb|alt=A photo of a complex wooden carving of animals, people and spirits laid on each other to about 2 meters in height|Complex wooden carving depicting images of power and daily life, such as horsemen, imported goods, military insignia, Europeans, rifles, wild beasts and masqueraders.]]'''Alusi''', also known as '''Arusi''' or '''Arushi''', are minor deities that are worshiped and served in Igbo mythology. There are a list of many different Alusi that exists within each community and each has its own purpose. When there is no longer need for the deity, it is returned to its source, through the help of a Chief Priest or Dibia, who is aware of the procedure and ensures that its done properly.

=== Mmuo ===
Mmuo simply means spirit. It is either a good and godly spirit (mmuo oma) or it is an evil spirit (mmuo ojo). For example, the Ogbanje spirit is seen as an evil spirit (mmuo ojo) and anyone possessed by this spirit is given spiritual attention. (Spiritual attention means a way of casting out the evil spirit through deliverance (Christian way) or through African Traditional Religion&amp;nbsp; (i.e. digging out his/her '''“iyi uwa”'''. the ATR way)). Ogbanje is an Igbo (Nigeria) term that means a repeater or someone who comes and departs.  Ogbanje is not a bad spirit in Igbo Cosmology. It is a word widely used to describe a kid or teenager who is claimed to die and be born repeatedly by the same person.

== Yam ==
The yam is very important to the Igbo as it is their staple crop. There are celebrations such as the New yam festival () which are held Every August of Every year for the harvesting of the yam.

The New Yam festival () is celebrated annually to secure a good harvest of the staple crop.  The festival is practiced primarily in Nigeria and other countries in West Africa.

=== Traditional marriage ===
Marriages in Igbo community follow a multi-step process before the bride and groom are proclaimed husband and wife in accordance with local law and tradition.
[[File:Igba nkwu ceremony 04.jpg|alt=Igbo Traditional Marriage|thumb|Traditional Igbo Marriage Attire]]
The traditional marriage is known as &quot;Igbankwu Alumdi&quot; in Igbo land, or wine carrying, since it involves the bride serving up a cup of palm wine to her fiancé. Prior to the wedding, the groom must go to the bride's compound with his father before the Igbankwu day to get the bride's father's consent to marry his daughter. If the bride's father is late, in this case, the bride's brother, uncle or male relative fills in for the bride's late father, as applies to the groom. On the second visit, when kola nuts (oji Igbo) are offered, the two fathers must arrange a price for the bride. In most cases, the bride's price is just symbolic, in addition to other requirements like kola nuts, goats, wine, fowl and so on. Normally, it takes more than one evening until the bride price is agreed upon, after which a feast is served to both parents. When the bride price is paid, another evening is set aside for the ceremony.

During the ceremony, the bride's father fills a cup with palm wine and hands it over to the daughter. Accompanied by her brides maids known as umuagbo nwunye, she then searches for the groom among the crowd of wedding guests to offer him the drink. Once the drink is offered, the bride and groom dance to the bride's father. They kneel before him and he will give them his blessings. After that, the couple dances for a while before taking their seats, then refreshment takes place followed by presentation of gifts, at times a speech from the MC, and then closing prayer and departure.

==Igbo Architecture==
Igbo architecture refers to the architectural styles and building traditions of the Igbo people. The architectural style is closely tied to the Igbo society's culture, beliefs, and social structure. While the architectural style has evolved, traditional Igbo architecture shares some common characteristics such as:

'''Compound layout'''- Igbo architectural traditions often revolve around the concept of a compound which is characterized by an enclosed area encompassing multiple family residences, open central courtyards, verandas, and auxiliary structures. These compounds are meticulously planned and sometimes paved with flat stones to foster communal living and facilitate familial engagements. Additionally, certain compounds feature unique elements like Impluvium houses, Gardens, Moats, and water wells demonstrating the diversity within Igbo architectural practices.

'''Ventilation''' - Igbo architecture integrates strategic placement of openings in buildings to promote cross-ventilation, aiding in regulating indoor temperatures. Employing expansive openings facilitates air circulation, ensuring occupant comfort. Depending on the area with high temperatures and humidity, evaporation of sweat becomes challenging; however, airflow aids this process, enhancing comfort. Moreover, construction practices involve thick walls, thatched roofs, and raised foundations to mitigate environmental challenges. The thick walls maintain cooler interiors in hot weather and warmth during rainy seasons. Thatched roofs provide insulation from direct sunlight, offering shade and contributing to thermal comfort.

'''Shrines and Sacred Spaces'''- Igbo architecture often includes designated spaces in compounds or community areas for ancestral shrines/temples and secret society meeting houses. These spaces are considered sacred and are an essential part of Igbo cultural and religious practices. These sacred structures may vary in design, ranging from simple open-air spaces to more elaborate structures with specific architectural features.

'''Decorative Elements -''' Traditional Igbo architecture often incorporates decorative elements, including painted designs on walls such as [[Uli (design)|uli]], carved wooden door frames, and intricate patterns on ceilings. These decorations may have symbolic or religious significance.

== Traditional attire ==
Igbo traditional attire varies across regions of Southeastern and south south Nigeria with various cultural significance. 

'''&lt;big&gt;Men&lt;/big&gt;'''

For men, common garments include ''uwe mwuda'' or ''afe ntutu'' ( robe) or ''efe elu'', a basic shirt paired with underneath wrappers or skirts complemented by the ''okpu ozo'' (the feathered red cap), or ''Okpu aji'' (woolen cap), ''ofo'', ''mkpara'' (staff) and ''Akupe'' (handfans) for ceremonial or titled occasions while  loin clothes or waist wrappers were usually worn as casual wears or basic activities like hunting or farming. 

&lt;big&gt;'''Women'''&lt;/big&gt;

[[File:Igbo_woman_wearing_Akwete_obiakwa(double_wrapper)_with_uweobi(blouse)_and_Ichafu_headdress.jpg|thumb|Igbo women’s traditional attire showing Obiakwa(matching double wrappers) made of Akwete George, uweobi (blouse with puffed sleeves) and stiff Ichafu (headdress)]]

Traditional Igbo women's attire comprises many regional and age-based (''Ụmụagbọ'') variant, including the Obiakwa pair of matching wrappers, Uweobi (blouse), and Ịchafụ̀ (head-tie), an elaborate and voluminous headdress traditionally worn by mature women. 

[[File:Igbo_woman_wearing_Isiagu_obiakwa_maiden_attire,_aka_olu(coral_beads)_ngala(head_beads)_and_nza(horsetail).jpg|thumb|A short Obiakwa (wrapper-style) ensemble paired with a fitted blouse, complemented by nza (horsewhisk), ngala (head beads), and other beaded accessories. The textile features  Isiagu motif]]

Younger women may wear shorter Obiakwa wrappers paired with a tubular Uweobi blouse. Traditional adornments include ''aka olu'' (coral beads), ''ngala'' (head beads), and ''mgbaji'' (waist beads), often complemented by other ceremonial accessories like the ''akupe'' (hand fans) and ''nza'' (horsetail whisks). Traditional attire and adornment form many parts of Igbo cultural expressions associated with age, status, ceremony, and identity.

'''&lt;u&gt;Obiakwa&lt;/u&gt;''' 

The ''Obiakwa'' is a traditional women's double- wrapper attire unique to Igbo weaving traditions such as Akwete cloth. It consists of a pair of matching wrappers Descriptions of Akwete weaving note that such wrapper sets were engineered during the weaving process to be worn together.  It is therefore sold in matching pairs.  These wrappers are standardly paired with a blouse called uweobi and the Ichafu headdress. Younger women usually wear shorter ''obiakwa'' waist wrapper sets combined with fitted or tubular blouses or wrappers . These clothings are also complemented with ''ngala'', ''mbaji'' and ''aka'' (beaded accessories) as well ''uli'' body arts. In some regions, The Uli body art was also used to decorate both men and women in the form of lines forming patterns and shapes on the body. 

'''&lt;u&gt;Blouse&lt;/u&gt;''' 

To complete the silhouette, double wrappers(obiakwa) are paired with a Blouse (or ''uweobi''), a traditional fitted blouse. Short ''obiakwa'' styles are usually paired with tubular blouses. 


'''&lt;u&gt;Ichafu&lt;/u&gt;'''

[[File:Igbo_woman_wearing_Joojii_and_Ichafu._Igbo_regality.jpg|thumb|An Igbo woman dressed in traditional attire consisting of a white puff-sleeve Uweobi blouse, a red and gold double George wrapper, and a stiff, elaborately structured Ichafu headdress, accessorized with pearl jewelry.]]

''Ichafu'' is an elaborate head-tie or headdress worn by Igbo women, especially for church services, ceremonies and other social occasions.  It forms part of a broader clothing ensemble that may include wrappers, blouses and jewellery, and is typically tied in volumnious elevated layered styles with large folds and pleats rising above the head. Ichafu is tied with various textiles including synthetic damask, brocade, Akwete and George fabric, which gives it the stiff and highly elaborated look. In Ogadinma, published by Granta, women were described wearing colourful blouses with “expensive ichafu” tied “in layers and pleats until the scarves were piled atop their heads like large plants”. Other dialectical variations for Ichafu is ''Akwaisi'', ''ulari'', ''unari'', ''nsu n'isi'', ''ufu isi'', ''asusu isi'', ''nchafu isi,'' ''Akishi''.   

'''&lt;u&gt;Textiles&lt;/u&gt;'''

Textiles commonly used across Igbo land include ''Isiagu'' (often patterned with the tiger or Lion head motifs), ''Akwete'' and ''Akwaocha'' handwoven clothes, and richly patterned George wrappers.

[[File:Little world, Aichi prefecture - African plaza - Hat of a vassal - Ìgbo people in Nigeria - Collected in 2006.jpg|110px|thumb|left|A traditional Igbo hat made entirely from [[wool]].]]Women carried their babies on their backs with a strip of clothing binding the two with a knot at her chest. This baby carrying technique was and still is practiced by many people groups across Africa, including the Igbo. This method has been modernized in the form of the child carrier.  Both men and women wore wrappers.[[File:Igba nkwu ceremony 07.jpg|thumb|Igba nkwu, Igbo traditional marriage]]

[[File:Igbo Traditional marriage.jpg|thumb|Igbo Traditional Marriage attire]]

==Chieftaincy Title==
[[File:Igbo ichi marks.jpg|thumb|An Igbo man with ''Ichi'' marks, a sign of rank as an Ozo]]
Highly accomplished men and women are admitted into their noble orders for people of title such as Ndi Ozo or Ndi Nze. These people receive insignia to show their stature. Membership is highly exclusive, and to qualify an individual need to be highly regarded and well-spoken of in the community.

== Apprenticeship ==
The Igbo have a unique form of apprenticeship in which either a male family member or a community member will spend time (usually in their teens to their adulthood) with another family, when they work for them. After the time spent with the family, the head of the host household, who is usually the older man who brought the apprentice into his household, will establish  the apprentice by either setting up a business for him or giving money or tools by which to make a living.

This practice was exploited by Europeans, who used this practice as a way of trading in enslaved people. Olaudah Equiano, although stolen from his home, was an Igbo person who was forced into service to an African family.  He said that he felt part of the family, unlike later, when he was shipped to North America and enslaved in the Thirteen Colonies.

The Igbo apprenticeship system is called Imu Ahia or Igba Boy in Igboland. It became more prominent among the Igbos after the Nigerian civil war, in a quest to survive the £20 policy which was proposed by Obafemi Awolowo that only £20 be given to every Biafran citizen to survive on regardless of what they had in the bank before the war and the rest of the money were held by the Nigerian government.

Petty trade was one of the only ways to build back destroyed communities as well as farming, but then, farming required time that was not readily available at that moment. Essentially, most people went into trading.

This Imu-Ahia/Igba Boy model was simple, it works in such a way that business owners would take in younger boys which can be relative, sibling or non-relative from same region, house them and have them work as apprentices in business while learning how it works and the secrets of the business. After the allotted time for the training was reached, 5–8 years’ time, a little graduation ceremony would be held for the '''Nwa Boy''' (the person that learnt the trade). He would also be paid a lump sum for their services over the years, and the money will be used to start a business for the '''Nwa Boy'''.

== Osu caste system ==
Osu are a group of people whose ancestors were dedicated to serving in shrines and temples for the deities of the Igbo, and therefore were deemed property of the gods. Relationships and sometimes interactions with Osu were (and to this day, still are) in many cases, forbidden.

To this day being called an ''Osu'' remains a stigma that prevents people's progress and lifestyles.

== Calendar (Iguafo Igbo) ==
In the traditional Igbo calendar, a week has 4 days  (''Eke'', ''Orie'', ''Afọ'', ''Nkwọ''), seven weeks make one month, a month has 28 days and there are 13 months in a year. In the last month, an extra day is added. The names of the days have their roots in the mythology of the Kingdom of Nri. It was believed that Eri, the sky-born founder of the Nri kingdom, had gone on a journey to discover the mystery of time. On his journey he had saluted and counted the four days by the names of the spirits that governed them, and so the names of the spirits (''eke'', ''orie'', ''afọ'' and ''Nkwo'') became the days of the week.

{{col-begin}}{{col-2}}
{| class=&quot;wikitable&quot;
!No. || Months (Ọnwa) || Gregorian equivalent
|-
|1 || '''Ọnwa Mbụ''' || (3rd week of February)
|-
|2 || '''Ọnwa Abụa''' || (March)
|-
|3 || '''Ọnwa Ife Eke''' || (April)
|-
|4 || '''Ọnwa Anọ''' || (May)
|-
|5 || '''Ọnwa Agwụ''' || (June)
|-
|6 || '''Ọnwa Ifejiọkụ''' || (July)
|-
|7 || '''Ọnwa Alọm Chi''' || (August to early September)
|-
|8 || '''Ọnwa Ilo Mmụọ''' || (Late September)
|-
|9 || '''Ọnwa Ana''' || (October)
|-
|10 || '''Ọnwa Okike''' || (Early November)
|-
|11 || '''Ọnwa Ajana''' || (Late November)
|-
|12 || '''Ọnwa Ede Ajana''' || (Late November to December)
|-
|13 || '''Ọnwa Ụzọ Alụsị''' || (January to early February)&lt;ref&gt;{{cite book |last1=Onwuejeogwu |first1=M. Angulu |title=An Igbo Civilization: Nri Kingdom &amp; Hegemony |date=1981 |publisher=Ethnographica |isbn=978-978-123-105-6 }}{{page needed|date=January 2024}}&lt;/ref&gt;&lt;ref&gt;{{cite web |url=http://www.free-press-release.com/news/200802/1204305180.html |title=Eze Nri - Igu-Aro Festival - 1008th AD |publisher=Free-Press-Release Inc. |date=February 29, 2008 |access-date=2010-04-06 |archive-date=2009-07-24 |archive-url=https://web.archive.org/web/20090724025818/http://www.free-press-release.com/news/200802/1204305180.html |url-status=dead }}&lt;/ref&gt;
|}

{{col-break}}
An example of a month: '''''Ọnwa Mbụ'''''

{| class=&quot;wikitable&quot;
|-
!Eke || Orie || Afọ || Nkwọ
|-
||||| 1 || 2
|-
|3 || 4 || 5 || 6
|-
|7 || 8 || 9 || 10
|-
|11 || 12 || 13 || 14
|-
|15 || 16 || 17 || 18
|-
|19 || 20 || 21 || 22
|-
|23 || 24 || 25 || 26
|-
|27 || 28||||
|}
{{col-end}}

=== Naming after market days ===
Newborn babies were sometimes named after the day of the week when born. This is no longer the fashion.  Names such as ''Mgbeke'' (maiden [born] on the day of Eke), Mgborie (maiden [born] on the Orie day) are commonly seen among the Igbo people. For males, ''Mgbe'' is replaced by ''Nwa'' or &quot;Okoro&quot; (Igbo: Child [of]). Examples of this are Solomon Okoronkwo and Nwankwo Kanu, two popular footballers.

== Igbo masks and masquerades ==
There are two basic types of masquerades, visible and invisible. The visible masquerades are meant for the public. They often are more entertaining. Masks used offer a visual appeal for their shapes and forms. In these visible masquerades, performances of harassment, music, dance, and parodies are acted out (Oyeneke 25).

The invisible masquerades take place at night.  Sound is the main tool for them. The masquerader uses his voice to scream so it may be heard throughout the village. The masks used are usually fierce looking and their interpretation is only fully understood by the society's members.  These invisible masquerades call upon a silent village to strike fear in the hearts of those not initiated into their society.

== Kola nut (Ọjị) ==
[[File:Kola nut.jpg|alt=Kola nut|thumb|Kola nut]]
Kola nut  occupies a unique position in the cultural life of Igbo people. Ọjị is the first thing served to any visitor in an Igbo home. Ọjị is served before an important function begins, be it marriage ceremony, settlement of family disputes or entering into any type of agreement. Ọjị is traditionally broken into pieces by hand, and if the Kola nut breaks into 3 pieces a special celebration is arranged.

[[Category:Igbo culture| ]]
[[Category:Culture of Nigeria]]
[[Category:Culture of Africa by ethnicity]]
[[Category:Igbo people]]
[[Category:Igbo society]]</text>
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      <text bytes="21081" sha1="3hz6p09agdmkvgfkx7p9zmqgr0n9hqu" xml:space="preserve">{{Short description|Cultural traditions of the Igbo people}}
'''Igbo culture''' are the customs, practices and traditions of the Igbo people of southeastern Nigeria. It consists of ancient practices known as ''Odinala'' ''ndi'' ''igbo'' as well as new concepts added into the Igbo culture either by cultural evolution or by outside influence. These customs and traditions includes the Igbo people's visual art, music and dance forms, as well as their attire, Food, cuisine and language dialects. Because of their various subgroups, the variety of their culture is heightened further.

== Music ==
[[File:Udu.jpg|thumb|right|95px|Udu, an Igbo instrument]]
The Igbo people have a melodic and symphonic musical style. Instruments include Ọ̀pì  otherwise known as '''Oja''' a wind instrument similar to the flute, '''igba''', and '''ichaka'''.

Another popular musical form among Igbo people is highlife, which is a fusion of jazz and traditional music and widely popular in West Africa. The modern Igbo highlife is seen in the works of Prince Nico Mbarga, Dr Sir Warrior, Oliver De Coque, Bright Chimezie, Celestine Ukwu,Chief Osita Osadebe, And many others who are some of the greatest Igbo highlife musicians of the twentieth century. There are also other notable Igbo highlife artists, like the Mike Ejeagha, Paulson Kalu, Ali Chukwuma, Ozoemena Nwa Nsugbe.

== Art ==
Igbo art is known for various types of masquerades, masks, outfits (symbolizing people), animals and abstract conceptions. Igbo art is also known for its bronze castings found in the town of Igbo Ukwu from the 9th century.

&lt;gallery widths=&quot;200&quot; heights=&quot;200&quot; mode=&quot;packed&quot;&gt;
File:Nigeria, igbo, maschera-elmo della società mmuo, xx secolo.jpg|Helmet-mask; 20th century; Indianapolis Museum of Art (USA)
File:Nigeria, igbo, figura femminile per un tempietto, xx secolo.jpg|Female figure for a small temple, 20th century; Indianapolis Museum of Art
File:Igbo brass anklet.jpg|Anklet beaten from a solid brass bar of the type worn by Igbo women. Now in the collection of Wolverhampton Art Gallery. The leg-tube extends approximately 7&amp;nbsp;cm each side of the 35&amp;nbsp;cm disc.
File:Bronze ceremonial vessel in form of a snail shell, 9th century, Igbo-Ukwu, Nigeria.JPG|Bronze ceremonial vessel in form of a snail shell; 9th century; from Igbo-Ukwu; Nigerian National Museum (Lagos, Nigeria)
File:Eze Onyiudo (2).jpg|Eze Onyiudo Masquerade Awka-Etiti
&lt;/gallery&gt;

== Mythology ==
While today many Igbo people are Christian, the traditional ancient Igbo religion is known as Odinani. In the Igbo mythology, which is part of their ancient religion, the supreme God is called Chineke (&quot;the God of creation&quot;); Chineke created the world and everything in it and is associated with all things on Earth. To the ancient Igbo, the cosmos is divided into four complex parts:
 
* OKIKE (Creation)
* ALUSI (Supernatural Forces or Deities)
* MMUO (Spirit)
* UWA (World)

=== Alusi ===
[[File:Complex sculpture Nigeria BM Af1954 23 522 img02.jpg|thumb|alt=A photo of a complex wooden carving of animals, people and spirits laid on each other to about 2 meters in height|Complex wooden carving depicting images of power and daily life, such as horsemen, imported goods, military insignia, Europeans, rifles, wild beasts and masqueraders.]]'''Alusi''', also known as '''Arusi''' or '''Arushi''', are minor deities that are worshiped and served in Igbo mythology. There are a list of many different Alusi that exists within each community and each has its own purpose. When there is no longer need for the deity, it is returned to its source, through the help of a Chief Priest or Dibia, who is aware of the procedure and ensures that its done properly.

=== Mmuo ===
Mmuo simply means spirit. It is either a good and godly spirit (mmuo oma) or it is an evil spirit (mmuo ojo). For example, the Ogbanje spirit is seen as an evil spirit (mmuo ojo) and anyone possessed by this spirit is given spiritual attention. (Spiritual attention means a way of casting out the evil spirit through deliverance (Christian way) or through African Traditional Religion&amp;nbsp; (i.e. digging out his/her '''“iyi uwa”'''. the ATR way)). Ogbanje is an Igbo (Nigeria) term that means a repeater or someone who comes and departs.  Ogbanje is not a bad spirit in Igbo Cosmology. It is a word widely used to describe a kid or teenager who is claimed to die and be born repeatedly by the same person.

== Yam ==
The yam is very important to the Igbo as it is their staple crop. There are celebrations such as the New yam festival  which are held Every August of Every year for the harvesting of the yam.

The New Yam festival  is celebrated annually to secure a good harvest of the staple crop.  The festival is practiced primarily in Nigeria and other countries in West Africa.

=== Traditional marriage ===
Marriages in Igbo community follow a multi-step process before the bride and groom are proclaimed husband and wife in accordance with local law and tradition.
[[File:Igba nkwu ceremony 04.jpg|alt=Igbo Traditional Marriage|thumb|Traditional Igbo Marriage Attire]]
The traditional marriage is known as &quot;Igbankwu Alumdi&quot; in Igbo land, or wine carrying, since it involves the bride serving up a cup of palm wine to her fiancé. Prior to the wedding, the groom must go to the bride's compound with his father before the Igbankwu day to get the bride's father's consent to marry his daughter. If the bride's father is late, in this case, the bride's brother, uncle or male relative fills in for the bride's late father, as applies to the groom. On the second visit, when kola nuts (oji Igbo) are offered, the two fathers must arrange a price for the bride. In most cases, the bride's price is just symbolic, in addition to other requirements like kola nuts, goats, wine, fowl and so on. Normally, it takes more than one evening until the bride price is agreed upon, after which a feast is served to both parents. When the bride price is paid, another evening is set aside for the ceremony.

During the ceremony, the bride's father fills a cup with palm wine and hands it over to the daughter. Accompanied by her brides maids known as umuagbo nwunye, she then searches for the groom among the crowd of wedding guests to offer him the drink. Once the drink is offered, the bride and groom dance to the bride's father. They kneel before him and he will give them his blessings. After that, the couple dances for a while before taking their seats, then refreshment takes place followed by presentation of gifts, at times a speech from the MC, and then closing prayer and departure.

==Igbo Architecture==
Igbo architecture refers to the architectural styles and building traditions of the Igbo people. The architectural style is closely tied to the Igbo society's culture, beliefs, and social structure. While the architectural style has evolved, traditional Igbo architecture shares some common characteristics such as:

'''Compound layout'''- Igbo architectural traditions often revolve around the concept of a compound which is characterized by an enclosed area encompassing multiple family residences, open central courtyards, verandas, and auxiliary structures. These compounds are meticulously planned and sometimes paved with flat stones to foster communal living and facilitate familial engagements. Additionally, certain compounds feature unique elements like Impluvium houses, Gardens, Moats, and water wells demonstrating the diversity within Igbo architectural practices.

'''Ventilation''' - Igbo architecture integrates strategic placement of openings in buildings to promote cross-ventilation, aiding in regulating indoor temperatures. Employing expansive openings facilitates air circulation, ensuring occupant comfort. Depending on the area with high temperatures and humidity, evaporation of sweat becomes challenging; however, airflow aids this process, enhancing comfort. Moreover, construction practices involve thick walls, thatched roofs, and raised foundations to mitigate environmental challenges. The thick walls maintain cooler interiors in hot weather and warmth during rainy seasons. Thatched roofs provide insulation from direct sunlight, offering shade and contributing to thermal comfort.

'''Shrines and Sacred Spaces'''- Igbo architecture often includes designated spaces in compounds or community areas for ancestral shrines/temples and secret society meeting houses. These spaces are considered sacred and are an essential part of Igbo cultural and religious practices. These sacred structures may vary in design, ranging from simple open-air spaces to more elaborate structures with specific architectural features.

'''Decorative Elements -''' Traditional Igbo architecture often incorporates decorative elements, including painted designs on walls such as [[Uli (design)|uli]], carved wooden door frames, and intricate patterns on ceilings. These decorations may have symbolic or religious significance.

== Traditional attire ==
Igbo traditional attire varies across regions of Southeastern and south south Nigeria with various cultural significance. 

'''&lt;big&gt;Men&lt;/big&gt;'''

For men, common garments include ''uwe mwuda'' or ''afe ntutu'' ( robe) or ''efe elu'', a basic shirt paired with underneath wrappers or skirts complemented by the ''okpu ozo'' (the feathered red cap), or ''Okpu aji'' (woolen cap), ''ofo'', ''mkpara'' (staff) and ''Akupe'' (handfans) for ceremonial or titled occasions while  loin clothes or waist wrappers were usually worn as casual wears or basic activities like hunting or farming. 

&lt;big&gt;'''Women'''&lt;/big&gt;

[[File:Igbo_woman_wearing_Akwete_obiakwa(double_wrapper)_with_uweobi(blouse)_and_Ichafu_headdress.jpg|thumb|Igbo women’s traditional attire showing Obiakwa(matching double wrappers) made of Akwete George, uweobi (blouse with puffed sleeves) and stiff Ichafu (headdress)]]

Traditional Igbo women's attire comprises many regional and age-based (''Ụmụagbọ'') variant, including the Obiakwa pair of matching wrappers, Uweobi (blouse), and Ịchafụ̀ (head-tie), an elaborate and voluminous headdress traditionally worn by mature women. 

[[File:Igbo_woman_wearing_Isiagu_obiakwa_maiden_attire,_aka_olu(coral_beads)_ngala(head_beads)_and_nza(horsetail).jpg|thumb|A short Obiakwa (wrapper-style) ensemble paired with a fitted blouse, complemented by nza (horsewhisk), ngala (head beads), and other beaded accessories. The textile features  Isiagu motif]]

Younger women may wear shorter Obiakwa wrappers paired with a tubular Uweobi blouse. Traditional adornments include ''aka olu'' (coral beads), ''ngala'' (head beads), and ''mgbaji'' (waist beads), often complemented by other ceremonial accessories like the ''akupe'' (hand fans) and ''nza'' (horsetail whisks). Traditional attire and adornment form many parts of Igbo cultural expressions associated with age, status, ceremony, and identity.

'''&lt;u&gt;Obiakwa&lt;/u&gt;''' 

The ''Obiakwa'' is a traditional women's double- wrapper attire unique to Igbo weaving traditions such as Akwete cloth. It consists of a pair of matching wrappers Descriptions of Akwete weaving note that such wrapper sets were engineered during the weaving process to be worn together.  It is therefore sold in matching pairs.  These wrappers are standardly paired with a blouse called uweobi and the Ichafu headdress. Younger women usually wear shorter ''obiakwa'' waist wrapper sets combined with fitted or tubular blouses or wrappers . These clothings are also complemented with ''ngala'', ''mbaji'' and ''aka'' (beaded accessories) as well ''uli'' body arts. In some regions, The Uli body art was also used to decorate both men and women in the form of lines forming patterns and shapes on the body. 

'''&lt;u&gt;Blouse&lt;/u&gt;''' 

To complete the silhouette, double wrappers(obiakwa) are paired with a Blouse (or ''uweobi''), a traditional fitted blouse. Short ''obiakwa'' styles are usually paired with tubular blouses. 


'''&lt;u&gt;Ichafu&lt;/u&gt;'''

[[File:Igbo_woman_wearing_Joojii_and_Ichafu._Igbo_regality.jpg|thumb|An Igbo woman dressed in traditional attire consisting of a white puff-sleeve Uweobi blouse, a red and gold double George wrapper, and a stiff, elaborately structured Ichafu headdress, accessorized with pearl jewelry.]]

''Ichafu'' is an elaborate head-tie or headdress worn by Igbo women, especially for church services, ceremonies and other social occasions.  It forms part of a broader clothing ensemble that may include wrappers, blouses and jewellery, and is typically tied in volumnious elevated layered styles with large folds and pleats rising above the head. Ichafu is tied with various textiles including synthetic damask, brocade, Akwete and George fabric, which gives it the stiff and highly elaborated look. In Ogadinma, published by Granta, women were described wearing colourful blouses with “expensive ichafu” tied “in layers and pleats until the scarves were piled atop their heads like large plants”. Other dialectical variations for Ichafu is ''Akwaisi'', ''ulari'', ''unari'', ''nsu n'isi'', ''ufu isi'', ''asusu isi'', ''nchafu isi,'' ''Akishi''.   

'''&lt;u&gt;Textiles&lt;/u&gt;'''

Textiles commonly used across Igbo land include ''Isiagu'' (often patterned with the tiger or Lion head motifs), ''Akwete'' and ''Akwaocha'' handwoven clothes, and richly patterned George wrappers.

[[File:Little world, Aichi prefecture - African plaza - Hat of a vassal - Ìgbo people in Nigeria - Collected in 2006.jpg|110px|thumb|left|A traditional Igbo hat made entirely from [[wool]].]]Women carried their babies on their backs with a strip of clothing binding the two with a knot at her chest. This baby carrying technique was and still is practiced by many people groups across Africa, including the Igbo. This method has been modernized in the form of the child carrier.  Both men and women wore wrappers.[[File:Igba nkwu ceremony 07.jpg|thumb|Igba nkwu, Igbo traditional marriage]]

[[File:Igbo Traditional marriage.jpg|thumb|Igbo Traditional Marriage attire]]

==Chieftaincy Title==
[[File:Igbo ichi marks.jpg|thumb|An Igbo man with ''Ichi'' marks, a sign of rank as an Ozo]]
Highly accomplished men and women are admitted into their noble orders for people of title such as Ndi Ozo or Ndi Nze. These people receive insignia to show their stature. Membership is highly exclusive, and to qualify an individual need to be highly regarded and well-spoken of in the community.

== Apprenticeship ==
The Igbo have a unique form of apprenticeship in which either a male family member or a community member will spend time (usually in their teens to their adulthood) with another family, when they work for them. After the time spent with the family, the head of the host household, who is usually the older man who brought the apprentice into his household, will establish  the apprentice by either setting up a business for him or giving money or tools by which to make a living.

This practice was exploited by Europeans, who used this practice as a way of trading in enslaved people. Olaudah Equiano, although stolen from his home, was an Igbo person who was forced into service to an African family.  He said that he felt part of the family, unlike later, when he was shipped to North America and enslaved in the Thirteen Colonies.

The Igbo apprenticeship system is called Imu Ahia or Igba Boy in Igboland. It became more prominent among the Igbos after the Nigerian civil war, in a quest to survive the £20 policy which was proposed by Obafemi Awolowo that only £20 be given to every Biafran citizen to survive on regardless of what they had in the bank before the war and the rest of the money were held by the Nigerian government.

Petty trade was one of the only ways to build back destroyed communities as well as farming, but then, farming required time that was not readily available at that moment. Essentially, most people went into trading.

This Imu-Ahia/Igba Boy model was simple, it works in such a way that business owners would take in younger boys which can be relative, sibling or non-relative from same region, house them and have them work as apprentices in business while learning how it works and the secrets of the business. After the allotted time for the training was reached, 5–8 years’ time, a little graduation ceremony would be held for the '''Nwa Boy''' (the person that learnt the trade). He would also be paid a lump sum for their services over the years, and the money will be used to start a business for the '''Nwa Boy'''.

== Osu caste system ==
Osu are a group of people whose ancestors were dedicated to serving in shrines and temples for the deities of the Igbo, and therefore were deemed property of the gods. Relationships and sometimes interactions with Osu were (and to this day, still are) in many cases, forbidden.

To this day being called an ''Osu'' remains a stigma that prevents people's progress and lifestyles.

== Calendar (Iguafo Igbo) ==
In the traditional Igbo calendar, a week has 4 days  (''Eke'', ''Orie'', ''Afọ'', ''Nkwọ''), seven weeks make one month, a month has 28 days and there are 13 months in a year. In the last month, an extra day is added. The names of the days have their roots in the mythology of the Kingdom of Nri. It was believed that Eri, the sky-born founder of the Nri kingdom, had gone on a journey to discover the mystery of time. On his journey he had saluted and counted the four days by the names of the spirits that governed them, and so the names of the spirits (''eke'', ''orie'', ''afọ'' and ''Nkwo'') became the days of the week.

{{col-begin}}{{col-2}}
{| class=&quot;wikitable&quot;
!No. || Months (Ọnwa) || Gregorian equivalent
|-
|1 || '''Ọnwa Mbụ''' || (3rd week of February)
|-
|2 || '''Ọnwa Abụa''' || (March)
|-
|3 || '''Ọnwa Ife Eke''' || (April)
|-
|4 || '''Ọnwa Anọ''' || (May)
|-
|5 || '''Ọnwa Agwụ''' || (June)
|-
|6 || '''Ọnwa Ifejiọkụ''' || (July)
|-
|7 || '''Ọnwa Alọm Chi''' || (August to early September)
|-
|8 || '''Ọnwa Ilo Mmụọ''' || (Late September)
|-
|9 || '''Ọnwa Ana''' || (October)
|-
|10 || '''Ọnwa Okike''' || (Early November)
|-
|11 || '''Ọnwa Ajana''' || (Late November)
|-
|12 || '''Ọnwa Ede Ajana''' || (Late November to December)
|-
|13 || '''Ọnwa Ụzọ Alụsị''' || (January to early February)&lt;ref&gt;{{cite book |last1=Onwuejeogwu |first1=M. Angulu |title=An Igbo Civilization: Nri Kingdom &amp; Hegemony |date=1981 |publisher=Ethnographica |isbn=978-978-123-105-6 }}{{page needed|date=January 2024}}&lt;/ref&gt;&lt;ref&gt;{{cite web |url=http://www.free-press-release.com/news/200802/1204305180.html |title=Eze Nri - Igu-Aro Festival - 1008th AD |publisher=Free-Press-Release Inc. |date=February 29, 2008 |access-date=2010-04-06 |archive-date=2009-07-24 |archive-url=https://web.archive.org/web/20090724025818/http://www.free-press-release.com/news/200802/1204305180.html |url-status=dead }}&lt;/ref&gt;
|}

{{col-break}}
An example of a month: '''''Ọnwa Mbụ'''''

{| class=&quot;wikitable&quot;
|-
!Eke || Orie || Afọ || Nkwọ
|-
||||| 1 || 2
|-
|3 || 4 || 5 || 6
|-
|7 || 8 || 9 || 10
|-
|11 || 12 || 13 || 14
|-
|15 || 16 || 17 || 18
|-
|19 || 20 || 21 || 22
|-
|23 || 24 || 25 || 26
|-
|27 || 28||||
|}
{{col-end}}

=== Naming after market days ===
Newborn babies were sometimes named after the day of the week when born. This is no longer the fashion.  Names such as ''Mgbeke'' (maiden [born] on the day of Eke), Mgborie (maiden [born] on the Orie day) are commonly seen among the Igbo people. For males, ''Mgbe'' is replaced by ''Nwa'' or &quot;Okoro&quot; (Igbo: Child [of]). Examples of this are Solomon Okoronkwo and Nwankwo Kanu, two popular footballers.

== Igbo masks and masquerades ==
There are two basic types of masquerades, visible and invisible. The visible masquerades are meant for the public. They often are more entertaining. Masks used offer a visual appeal for their shapes and forms. In these visible masquerades, performances of harassment, music, dance, and parodies are acted out (Oyeneke 25).

The invisible masquerades take place at night.  Sound is the main tool for them. The masquerader uses his voice to scream so it may be heard throughout the village. The masks used are usually fierce looking and their interpretation is only fully understood by the society's members.  These invisible masquerades call upon a silent village to strike fear in the hearts of those not initiated into their society.

== Kola nut (Ọjị) ==
[[File:Kola nut.jpg|alt=Kola nut|thumb|Kola nut]]
Kola nut  occupies a unique position in the cultural life of Igbo people. Ọjị is the first thing served to any visitor in an Igbo home. Ọjị is served before an important function begins, be it marriage ceremony, settlement of family disputes or entering into any type of agreement. Ọjị is traditionally broken into pieces by hand, and if the Kola nut breaks into 3 pieces a special celebration is arranged.

[[Category:Igbo culture| ]]
[[Category:Culture of Nigeria]]
[[Category:Culture of Africa by ethnicity]]
[[Category:Igbo people]]
[[Category:Igbo society]]</text>
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      <text bytes="20925" sha1="9jjrbwj9ns1pllkq52d6k29clwnqrpz" xml:space="preserve">{{Short description|Cultural traditions of the Igbo people}}
'''Igbo culture''' are the customs, practices and traditions of the Igbo people of southeastern Nigeria. It consists of ancient practices known as ''Odinala'' ''ndi'' ''igbo'' as well as new concepts added into the Igbo culture either by cultural evolution or by outside influence. These customs and traditions includes the Igbo people's visual art, music and dance forms, as well as their attire, Food, cuisine and language dialects. Because of their various subgroups, the variety of their culture is heightened further.

== Music ==
[[File:Udu.jpg|thumb|right|95px|Udu, an Igbo instrument]]
The Igbo people have a melodic and symphonic musical style. Instruments include Ọ̀pì  otherwise known as '''Oja''' a wind instrument similar to the flute, '''igba''', and '''ichaka'''.

Another popular musical form among Igbo people is highlife, which is a fusion of jazz and traditional music and widely popular in West Africa. The modern Igbo highlife is seen in the works of Prince Nico Mbarga, Dr Sir Warrior, Oliver De Coque, Bright Chimezie, Celestine Ukwu,Chief Osita Osadebe, And many others who are some of the greatest Igbo highlife musicians of the twentieth century. There are also other notable Igbo highlife artists, like the Mike Ejeagha, Paulson Kalu, Ali Chukwuma, Ozoemena Nwa Nsugbe.

== Art ==
Igbo art is known for various types of masquerades, masks, outfits (symbolizing people), animals and abstract conceptions. Igbo art is also known for its bronze castings found in the town of Igbo Ukwu from the 9th century.

&lt;gallery widths=&quot;200&quot; heights=&quot;200&quot; mode=&quot;packed&quot;&gt;
File:Nigeria, igbo, maschera-elmo della società mmuo, xx secolo.jpg|Helmet-mask; 20th century; Indianapolis Museum of Art (USA)
File:Nigeria, igbo, figura femminile per un tempietto, xx secolo.jpg|Female figure for a small temple, 20th century; Indianapolis Museum of Art
File:Igbo brass anklet.jpg|Anklet beaten from a solid brass bar of the type worn by Igbo women. Now in the collection of Wolverhampton Art Gallery. The leg-tube extends approximately 7&amp;nbsp;cm each side of the 35&amp;nbsp;cm disc.
File:Bronze ceremonial vessel in form of a snail shell, 9th century, Igbo-Ukwu, Nigeria.JPG|Bronze ceremonial vessel in form of a snail shell; 9th century; from Igbo-Ukwu; Nigerian National Museum (Lagos, Nigeria)
File:Eze Onyiudo (2).jpg|Eze Onyiudo Masquerade Awka-Etiti
&lt;/gallery&gt;

== Mythology ==
While today many Igbo people are Christian, the traditional ancient Igbo religion is known as Odinani. In the Igbo mythology, which is part of their ancient religion, the supreme God is called Chineke (&quot;the God of creation&quot;); Chineke created the world and everything in it and is associated with all things on Earth. To the ancient Igbo, the cosmos is divided into four complex parts:
 
* OKIKE (Creation)
* ALUSI (Supernatural Forces or Deities)
* MMUO (Spirit)
* UWA (World)

=== Alusi ===
[[File:Complex sculpture Nigeria BM Af1954 23 522 img02.jpg|thumb|alt=A photo of a complex wooden carving of animals, people and spirits laid on each other to about 2 meters in height|Complex wooden carving depicting images of power and daily life, such as horsemen, imported goods, military insignia, Europeans, rifles, wild beasts and masqueraders.]]'''Alusi''', also known as '''Arusi''' or '''Arushi''', are minor deities that are worshiped and served in Igbo mythology. There are a list of many different Alusi that exists within each community and each has its own purpose. When there is no longer need for the deity, it is returned to its source, through the help of a Chief Priest or Dibia, who is aware of the procedure and ensures that its done properly.

=== Mmuo ===
Mmuo simply means spirit. It is either a good and godly spirit (mmuo oma) or it is an evil spirit (mmuo ojo). For example, the Ogbanje spirit is seen as an evil spirit (mmuo ojo) and anyone possessed by this spirit is given spiritual attention. (Spiritual attention means a way of casting out the evil spirit through deliverance (Christian way) or through African Traditional Religion&amp;nbsp; (i.e. digging out his/her '''“iyi uwa”'''. the ATR way)). Ogbanje is an Igbo (Nigeria) term that means a repeater or someone who comes and departs.  Ogbanje is not a bad spirit in Igbo Cosmology. It is a word widely used to describe a kid or teenager who is claimed to die and be born repeatedly by the same person.

== Yam ==
The yam is very important to the Igbo as it is their staple crop. There are celebrations such as the New yam festival  which are held Every August of Every year for the harvesting of the yam.

The New Yam festival  is celebrated annually to secure a good harvest of the staple crop.  The festival is practiced primarily in Nigeria and other countries in West Africa.

=== Traditional marriage ===
Marriages in Igbo community follow a multi-step process before the bride and groom are proclaimed husband and wife in accordance with local law and tradition.
[[File:Igba nkwu ceremony 04.jpg|alt=Igbo Traditional Marriage|thumb|Traditional Igbo Marriage Attire]]
The traditional marriage is known as &quot;Igbankwu Alumdi&quot; in Igbo land, or wine carrying, since it involves the bride serving up a cup of palm wine to her fiancé. Prior to the wedding, the groom must go to the bride's compound with his father before the Igbankwu day to get the bride's father's consent to marry his daughter. If the bride's father is late, in this case, the bride's brother, uncle or male relative fills in for the bride's late father, as applies to the groom. On the second visit, when kola nuts (oji Igbo) are offered, the two fathers must arrange a price for the bride. In most cases, the bride's price is just symbolic, in addition to other requirements like kola nuts, goats, wine, fowl and so on. Normally, it takes more than one evening until the bride price is agreed upon, after which a feast is served to both parents. When the bride price is paid, another evening is set aside for the ceremony.

During the ceremony, the bride's father fills a cup with palm wine and hands it over to the daughter. Accompanied by her brides maids known as umuagbo nwunye, she then searches for the groom among the crowd of wedding guests to offer him the drink. Once the drink is offered, the bride and groom dance to the bride's father. They kneel before him and he will give them his blessings. After that, the couple dances for a while before taking their seats, then refreshment takes place followed by presentation of gifts, at times a speech from the MC, and then closing prayer and departure.

==Igbo Architecture==
Igbo architecture refers to the architectural styles and building traditions of the Igbo people. The architectural style is closely tied to the Igbo society's culture, beliefs, and social structure. While the architectural style has evolved, traditional Igbo architecture shares some common characteristics such as:

'''Compound layout'''- Igbo architectural traditions often revolve around the concept of a compound which is characterized by an enclosed area encompassing multiple family residences, open central courtyards, verandas, and auxiliary structures. These compounds are meticulously planned and sometimes paved with flat stones to foster communal living and facilitate familial engagements. Additionally, certain compounds feature unique elements like Impluvium houses, Gardens, Moats, and water wells demonstrating the diversity within Igbo architectural practices.

'''Ventilation''' - Igbo architecture integrates strategic placement of openings in buildings to promote cross-ventilation, aiding in regulating indoor temperatures. Employing expansive openings facilitates air circulation, ensuring occupant comfort. Depending on the area with high temperatures and humidity, evaporation of sweat becomes challenging; however, airflow aids this process, enhancing comfort. Moreover, construction practices involve thick walls, thatched roofs, and raised foundations to mitigate environmental challenges. The thick walls maintain cooler interiors in hot weather and warmth during rainy seasons. Thatched roofs provide insulation from direct sunlight, offering shade and contributing to thermal comfort.

'''Shrines and Sacred Spaces'''- Igbo architecture often includes designated spaces in compounds or community areas for ancestral shrines/temples and secret society meeting houses. These spaces are considered sacred and are an essential part of Igbo cultural and religious practices. These sacred structures may vary in design, ranging from simple open-air spaces to more elaborate structures with specific architectural features.

'''Decorative Elements -''' Traditional Igbo architecture often incorporates decorative elements, including painted designs on walls such as [[Uli (design)|uli]], carved wooden door frames, and intricate patterns on ceilings. These decorations may have symbolic or religious significance.

== Traditional attire ==
Igbo traditional attire varies across regions of Southeastern and south south Nigeria with various cultural significance. 

'''&lt;big&gt;Men&lt;/big&gt;'''

For men, common garments include ''uwe mwuda'' or ''afe ntutu'' ( robe) or ''efe elu'', a basic shirt paired with underneath wrappers or skirts complemented by the ''okpu ozo'' (the feathered red cap), or ''Okpu aji'' (woolen cap), ''ofo'', ''mkpara'' (staff) and ''Akupe'' (handfans) for ceremonial or titled occasions while  loin clothes or waist wrappers were usually worn as casual wears or basic activities like hunting or farming. 

&lt;big&gt;'''Women'''&lt;/big&gt;

[[File:Igbo_woman_wearing_Akwete_obiakwa(double_wrapper)_with_uweobi(blouse)_and_Ichafu_headdress.jpg|thumb|Igbo women’s traditional attire showing Obiakwa(matching double wrappers) made of Akwete George, uweobi (blouse with puffed sleeves) and stiff Ichafu (headdress)]]

Traditional Igbo women's attire comprises many regional and age-based (''Ụmụagbọ'') variant, including the Obiakwa pair of matching wrappers, Uweobi (blouse), and Ịchafụ̀ (head-tie), an elaborate and voluminous headdress traditionally worn by mature women. 

[[File:Igbo_woman_wearing_Isiagu_obiakwa_maiden_attire,_aka_olu(coral_beads)_ngala(head_beads)_and_nza(horsetail).jpg|thumb|A short Obiakwa (wrapper-style) ensemble paired with a fitted blouse, complemented by nza (horsewhisk), ngala (head beads), and other beaded accessories. The textile features  Isiagu motif]]

Younger women may wear shorter Obiakwa wrappers paired with a tubular Uweobi blouse. Traditional adornments include ''aka olu'' (coral beads), ''ngala'' (head beads), and ''mgbaji'' (waist beads), often complemented by other ceremonial accessories like the ''akupe'' (hand fans) and ''nza'' (horsetail whisks). Traditional attire and adornment form many parts of Igbo cultural expressions associated with age, status, ceremony, and identity.

'''&lt;u&gt;Obiakwa&lt;/u&gt;''' 

The ''Obiakwa'' is a traditional women's double- wrapper attire unique to Igbo weaving traditions such as Akwete cloth. It consists of a pair of matching wrappers Descriptions of Akwete weaving note that such wrapper sets were engineered during the weaving process to be worn together.  It is therefore sold in matching pairs.  These wrappers are standardly paired with a blouse called uweobi and the Ichafu headdress. Younger women usually wear shorter ''obiakwa'' waist wrapper sets combined with fitted or tubular blouses or wrappers . These clothings are also complemented with ''ngala'', ''mbaji'' and ''aka'' (beaded accessories) as well ''uli'' body arts. In some regions, The Uli body art was also used to decorate both men and women in the form of lines forming patterns and shapes on the body. 

'''&lt;u&gt;Blouse&lt;/u&gt;''' 

To complete the silhouette, double wrappers(obiakwa) are paired with a Blouse (or ''uweobi''), a traditional fitted blouse. Short ''obiakwa'' styles are usually paired with tubular blouses. 


'''&lt;u&gt;Ichafu&lt;/u&gt;'''

[[File:Igbo_woman_wearing_Joojii_and_Ichafu._Igbo_regality.jpg|thumb|An Igbo woman dressed in traditional attire consisting of a white puff-sleeve Uweobi blouse, a red and gold double George wrapper, and a stiff, elaborately structured Ichafu headdress, accessorized with pearl jewelry.]]

''Ichafu'' is an elaborate head-tie or headdress worn by Igbo women, especially for church services, ceremonies and other social occasions.  It forms part of a broader clothing ensemble that may include wrappers, blouses and jewellery, and is typically tied in volumnious elevated layered styles with large folds and pleats rising above the head. Ichafu is tied with various textiles including synthetic damask, brocade, Akwete and George fabric, which gives it the stiff and highly elaborated look. In Ogadinma, published by Granta, women were described wearing colourful blouses with “expensive ichafu” tied “in layers and pleats until the scarves were piled atop their heads like large plants”. Other dialectical variations for Ichafu is ''Akwaisi'', ''ulari'', ''unari'', ''nsu n'isi'', ''ufu isi'', ''asusu isi'', ''nchafu isi,'' ''Akishi''.   

'''&lt;u&gt;Textiles&lt;/u&gt;'''

Textiles commonly used across Igbo land include ''Isiagu'' (often patterned with the tiger or Lion head motifs), ''Akwete'' and ''Akwaocha'' handwoven clothes, and richly patterned George wrappers.

[[File:Little world, Aichi prefecture - African plaza - Hat of a vassal - Ìgbo people in Nigeria - Collected in 2006.jpg|110px|thumb|left|A traditional Igbo hat made entirely from [[wool]].]]Women carried their babies on their backs with a strip of clothing binding the two with a knot at her chest. This baby carrying technique was and still is practiced by many people groups across Africa, including the Igbo. This method has been modernized in the form of the child carrier.  Both men and women wore wrappers.[[File:Igba nkwu ceremony 07.jpg|thumb|Igba nkwu, Igbo traditional marriage]]

[[File:Igbo Traditional marriage.jpg|thumb|Igbo Traditional Marriage attire]]

==Chieftaincy Title==
[[File:Igbo ichi marks.jpg|thumb|An Igbo man with ''Ichi'' marks, a sign of rank as an Ozo]]
Highly accomplished men and women are admitted into their noble orders for people of title such as Ndi Ozo or Ndi Nze. These people receive insignia to show their stature. Membership is highly exclusive, and to qualify an individual need to be highly regarded and well-spoken of in the community.

== Apprenticeship ==
The Igbo have a unique form of apprenticeship in which either a male family member or a community member will spend time (usually in their teens to their adulthood) with another family, when they work for them. After the time spent with the family, the head of the host household, who is usually the older man who brought the apprentice into his household, will establish  the apprentice by either setting up a business for him or giving money or tools by which to make a living.

This practice was exploited by Europeans, who used this practice as a way of trading in enslaved people. Olaudah Equiano, although stolen from his home, was an Igbo person who was forced into service to an African family.  He said that he felt part of the family, unlike later, when he was shipped to North America and enslaved in the Thirteen Colonies.

The Igbo apprenticeship system is called Imu Ahia or Igba Boy in Igboland. It became more prominent among the Igbos after the Nigerian civil war, in a quest to survive the £20 policy which was proposed by Obafemi Awolowo that only £20 be given to every Biafran citizen to survive on regardless of what they had in the bank before the war and the rest of the money were held by the Nigerian government.

Petty trade was one of the only ways to build back destroyed communities as well as farming, but then, farming required time that was not readily available at that moment. Essentially, most people went into trading.

This Imu-Ahia/Igba Boy model was simple, it works in such a way that business owners would take in younger boys which can be relative, sibling or non-relative from same region, house them and have them work as apprentices in business while learning how it works and the secrets of the business. After the allotted time for the training was reached, 5–8 years’ time, a little graduation ceremony would be held for the '''Nwa Boy''' (the person that learnt the trade). He would also be paid a lump sum for their services over the years, and the money will be used to start a business for the '''Nwa Boy'''.

== Osu caste system ==
Osu are a group of people whose ancestors were dedicated to serving in shrines and temples for the deities of the Igbo, and therefore were deemed property of the gods. Relationships and sometimes interactions with Osu were (and to this day, still are) in many cases, forbidden.

To this day being called an ''Osu'' remains a stigma that prevents people's progress and lifestyles.

== Calendar (Iguafo Igbo) ==
In the traditional Igbo calendar, a week has 4 days  (''Eke'', ''Orie'', ''Afọ'', ''Nkwọ''), seven weeks make one month, a month has 28 days and there are 13 months in a year. In the last month, an extra day is added. The names of the days have their roots in the mythology of the Kingdom of Nri. It was believed that Eri, the sky-born founder of the Nri kingdom, had gone on a journey to discover the mystery of time. On his journey he had saluted and counted the four days by the names of the spirits that governed them, and so the names of the spirits (''eke'', ''orie'', ''afọ'' and ''Nkwo'') became the days of the week.

{{col-begin}}{{col-2}}
{| class=&quot;wikitable&quot;
!No. || Months (Ọnwa) || Gregorian equivalent
|-
|1 || '''Ọnwa Mbụ''' || (3rd week of February)
|-
|2 || '''Ọnwa Abụa''' || (March)
|-
|3 || '''Ọnwa Ife Eke''' || (April)
|-
|4 || '''Ọnwa Anọ''' || (May)
|-
|5 || '''Ọnwa Agwụ''' || (June)
|-
|6 || '''Ọnwa Ifejiọkụ''' || (July)
|-
|7 || '''Ọnwa Alọm Chi''' || (August to early September)
|-
|8 || '''Ọnwa Ilo Mmụọ''' || (Late September)
|-
|9 || '''Ọnwa Ana''' || (October)
|-
|10 || '''Ọnwa Okike''' || (Early November)
|-
|11 || '''Ọnwa Ajana''' || (Late November)
|-
|12 || '''Ọnwa Ede Ajana''' || (Late November to December)
|-
|13 || '''Ọnwa Ụzọ Alụsị''' || (January to early February)&lt;ref&gt;{{cite book |last1=Onwuejeogwu |first1=M. Angulu |title=An Igbo Civilization: Nri Kingdom &amp; Hegemony |date=1981 |publisher=Ethnographica |isbn=978-978-123-105-6 }}{{page needed|date=January 2024}}&lt;/ref&gt;&lt;ref&gt;{{cite web |url=http://www.free-press-release.com/news/200802/1204305180.html |title=Eze Nri - Igu-Aro Festival - 1008th AD |publisher=Free-Press-Release Inc. |date=February 29, 2008 |access-date=2010-04-06 |archive-date=2009-07-24 |archive-url=https://web.archive.org/web/20090724025818/http://www.free-press-release.com/news/200802/1204305180.html |url-status=dead }}&lt;/ref&gt;
|}

{{col-break}}
An example of a month: '''''Ọnwa Mbụ'''''

{| class=&quot;wikitable&quot;
|-
!Eke || Orie || Afọ || Nkwọ
|-
||||| 1 || 2
|-
|3 || 4 || 5 || 6
|-
|7 || 8 || 9 || 10
|-
|11 || 12 || 13 || 14
|-
|15 || 16 || 17 || 18
|-
|19 || 20 || 21 || 22
|-
|23 || 24 || 25 || 26
|-
|27 || 28||||
|}
{{col-end}}

=== Naming after market days ===
Newborn babies were sometimes named after the day of the week when born. This is no longer the fashion.  Names such as ''Mgbeke'' (maiden [born] on the day of Eke), Mgborie (maiden [born] on the Orie day) are commonly seen among the Igbo people. For males, ''Mgbe'' is replaced by ''Nwa'' or &quot;Okoro&quot; (Igbo: Child [of]). Examples of this are Solomon Okoronkwo and Nwankwo Kanu, two popular footballers.

== Igbo masks and masquerades ==
There are two basic types of masquerades, visible and invisible. The visible masquerades are meant for the public. They often are more entertaining. Masks used offer a visual appeal for their shapes and forms. In these visible masquerades, performances of harassment, music, dance, and parodies are acted out (Oyeneke 25).

The invisible masquerades take place at night.  Sound is the main tool for them. The masquerader uses his voice to scream so it may be heard throughout the village. The masks used are usually fierce looking and their interpretation is only fully understood by the society's members.  These invisible masquerades call upon a silent village to strike fear in the hearts of those not initiated into their society.

== Kola nut (Ọjị) ==
[[File:Kola nut.jpg|alt=Kola nut|thumb|Kola nut]]
Kola nut  occupies a unique position in the cultural life of Igbo people. Ọjị is the first thing served to any visitor in an Igbo home. Ọjị is served before an important function begins, be it marriage ceremony, settlement of family disputes or entering into any type of agreement. Ọjị is traditionally broken into pieces by hand, and if the Kola nut breaks into 3 pieces a special celebration is arranged.</text>
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      <text bytes="22357" sha1="5o4da27fsmktclxym9uq097lxsfpjr0" xml:space="preserve">== Introduction ==
'''Igbo culture''' are the customs, practices and traditions of the Igbo people of southeastern Nigeria. It consists of ancient practices known as ''Odinala'' ''ndi'' ''igbo'' as well as new concepts added into the Igbo culture either by cultural evolution or by outside influence. These customs and traditions includes the Igbo people's visual art, music and dance forms, as well as their attire, Food, cuisine and language dialects. Because of their various subgroups, the variety of their culture is heightened further.

== Learning Objectives ==
By reviewing this material, you should be able to:

* '''Identify the major traditional musical instruments and art forms of Ndị Igbo.'''
* '''Contrast the components of traditional Ndị Igbo cosmology.'''
* '''Detail the socio-economic functions of historical practices like traditional marriage, architecture, and the apprenticeship system.'''


== Module 1: Creative Arts and Expressive Traditions ==

=== Music ===
[[File:Udu.jpg|thumb|right|95px|Udu, an Igbo instrument]]
The Igbo people have a melodic and symphonic musical style. Instruments include Ọ̀pì  otherwise known as '''Oja''' a wind instrument similar to the flute, '''igba''', and '''ichaka'''.

Another popular musical form among Igbo people is highlife, which is a fusion of jazz and traditional music and widely popular in West Africa. The modern Igbo highlife is seen in the works of Prince Nico Mbarga, Dr Sir Warrior, Oliver De Coque, Bright Chimezie, Celestine Ukwu,Chief Osita Osadebe, And many others who are some of the greatest Igbo highlife musicians of the twentieth century. There are also other notable Igbo highlife artists, like the Mike Ejeagha, Paulson Kalu, Ali Chukwuma, Ozoemena Nwa Nsugbe.

=== Art ===
Igbo art is known for various types of masquerades, masks, outfits (symbolizing people), animals and abstract conceptions. Igbo art is also known for its bronze castings found in the town of Igbo Ukwu from the 9th century.

&lt;gallery widths=&quot;200&quot; heights=&quot;200&quot; mode=&quot;packed&quot;&gt;
File:Nigeria, igbo, maschera-elmo della società mmuo, xx secolo.jpg|Helmet-mask; 20th century; Indianapolis Museum of Art (USA)
File:Nigeria, igbo, figura femminile per un tempietto, xx secolo.jpg|Female figure for a small temple, 20th century; Indianapolis Museum of Art
File:Igbo brass anklet.jpg|Anklet beaten from a solid brass bar of the type worn by Igbo women. Now in the collection of Wolverhampton Art Gallery. The leg-tube extends approximately 7&amp;nbsp;cm each side of the 35&amp;nbsp;cm disc.
File:Bronze ceremonial vessel in form of a snail shell, 9th century, Igbo-Ukwu, Nigeria.JPG|Bronze ceremonial vessel in form of a snail shell; 9th century; from Igbo-Ukwu; Nigerian National Museum (Lagos, Nigeria)
File:Eze Onyiudo (2).jpg|Eze Onyiudo Masquerade Awka-Etiti
&lt;/gallery&gt;

=== Igbo masks and masquerades ===
There are two basic types of masquerades, visible and invisible. The visible masquerades are meant for the public. They often are more entertaining. Masks used offer a visual appeal for their shapes and forms. In these visible masquerades, performances of harassment, music, dance, and parodies are acted out (Oyeneke 25).

The invisible masquerades take place at night.  Sound is the main tool for them. The masquerader uses his voice to scream so it may be heard throughout the village. The masks used are usually fierce looking and their interpretation is only fully understood by the society's members.  These invisible masquerades call upon a silent village to strike fear in the hearts of those not initiated into their society.


== Module 2: Spiritual Beliefs and Cosmological Frameworks ==

=== Mythology ===
While today many Igbo people are Christian, the traditional ancient Igbo religion is known as Odinani. In the Igbo mythology, which is part of their ancient religion, the supreme God is called Chineke (&quot;the God of creation&quot;); Chineke created the world and everything in it and is associated with all things on Earth. To the ancient Igbo, the cosmos is divided into four complex parts:
 
* OKIKE (Creation)
* ALUSI (Supernatural Forces or Deities)
* MMUO (Spirit)
* UWA (World)

==== Alusi ====
[[File:Complex sculpture Nigeria BM Af1954 23 522 img02.jpg|thumb|alt=A photo of a complex wooden carving of animals, people and spirits laid on each other to about 2 meters in height|Complex wooden carving depicting images of power and daily life, such as horsemen, imported goods, military insignia, Europeans, rifles, wild beasts and masqueraders.]]'''Alusi''', also known as '''Arusi''' or '''Arushi''', are minor deities that are worshiped and served in Igbo mythology. There are a list of many different Alusi that exists within each community and each has its own purpose. When there is no longer need for the deity, it is returned to its source, through the help of a Chief Priest or Dibia, who is aware of the procedure and ensures that its done properly.

==== Mmuo ====
Mmuo simply means spirit. It is either a good and godly spirit (mmuo oma) or it is an evil spirit (mmuo ojo). For example, the Ogbanje spirit is seen as an evil spirit (mmuo ojo) and anyone possessed by this spirit is given spiritual attention. (Spiritual attention means a way of casting out the evil spirit through deliverance (Christian way) or through African Traditional Religion&amp;nbsp; (i.e. digging out his/her '''“iyi uwa”'''. the ATR way)). Ogbanje is an Igbo (Nigeria) term that means a repeater or someone who comes and departs.  Ogbanje is not a bad spirit in Igbo Cosmology. It is a word widely used to describe a kid or teenager who is claimed to die and be born repeatedly by the same person.

=== Osu caste system ===
Osu are a group of people whose ancestors were dedicated to serving in shrines and temples for the deities of the Igbo, and therefore were deemed property of the gods. Relationships and sometimes interactions with Osu were (and to this day, still are) in many cases, forbidden.

To this day being called an ''Osu'' remains a stigma that prevents people's progress and lifestyles.


== Module 3: Social Milestones and Economic Structures ==

=== Yam ===
The yam is very important to the Igbo as it is their staple crop. There are celebrations such as the New yam festival  which are held Every August of Every year for the harvesting of the yam.

The New Yam festival  is celebrated annually to secure a good harvest of the staple crop.  The festival is practiced primarily in Nigeria and other countries in West Africa.

=== Traditional marriage ===
Marriages in Igbo community follow a multi-step process before the bride and groom are proclaimed husband and wife in accordance with local law and tradition.
[[File:Igba nkwu ceremony 04.jpg|alt=Igbo Traditional Marriage|thumb|Traditional Igbo Marriage Attire]]
The traditional marriage is known as &quot;Igbankwu Alumdi&quot; in Igbo land, or wine carrying, since it involves the bride serving up a cup of palm wine to her fiancé. Prior to the wedding, the groom must go to the bride's compound with his father before the Igbankwu day to get the bride's father's consent to marry his daughter. If the bride's father is late, in this case, the bride's brother, uncle or male relative fills in for the bride's late father, as applies to the groom. On the second visit, when kola nuts (oji Igbo) are offered, the two fathers must arrange a price for the bride. In most cases, the bride's price is just symbolic, in addition to other requirements like kola nuts, goats, wine, fowl and so on. Normally, it takes more than one evening until the bride price is agreed upon, after which a feast is served to both parents. When the bride price is paid, another evening is set aside for the ceremony.

During the ceremony, the bride's father fills a cup with palm wine and hands it over to the daughter. Accompanied by her brides maids known as umuagbo nwunye, she then searches for the groom among the crowd of wedding guests to offer him the drink. Once the drink is offered, the bride and groom dance to the bride's father. They kneel before him and he will give them his blessings. After that, the couple dances for a while before taking their seats, then refreshment takes place followed by presentation of gifts, at times a speech from the MC, and then closing prayer and departure.

=== Apprenticeship ===
The Igbo have a unique form of apprenticeship in which either a male family member or a community member will spend time (usually in their teens to their adulthood) with another family, when they work for them. After the time spent with the family, the head of the host household, who is usually the older man who brought the apprentice into his household, will establish  the apprentice by either setting up a business for him or giving money or tools by which to make a living.

This practice was exploited by Europeans, who used this practice as a way of trading in enslaved people. Olaudah Equiano, although stolen from his home, was an Igbo person who was forced into service to an African family.  He said that he felt part of the family, unlike later, when he was shipped to North America and enslaved in the Thirteen Colonies.

The Igbo apprenticeship system is called Imu Ahia or Igba Boy in Igboland. It became more prominent among the Igbos after the Nigerian civil war, in a quest to survive the £20 policy which was proposed by Obafemi Awolowo that only £20 be given to every Biafran citizen to survive on regardless of what they had in the bank before the war and the rest of the money were held by the Nigerian government.

Petty trade was one of the only ways to build back destroyed communities as well as farming, but then, farming required time that was not readily available at that moment. Essentially, most people went into trading.

This Imu-Ahia/Igba Boy model was simple, it works in such a way that business owners would take in younger boys which can be relative, sibling or non-relative from same region, house them and have them work as apprentices in business while learning how it works and the secrets of the business. After the allotted time for the training was reached, 5–8 years’ time, a little graduation ceremony would be held for the '''Nwa Boy''' (the person that learnt the trade). He would also be paid a lump sum for their services over the years, and the money will be used to start a business for the '''Nwa Boy'''.

=== Chieftaincy Title ===
[[File:Igbo ichi marks.jpg|thumb|An Igbo man with ''Ichi'' marks, a sign of rank as an Ozo]]
Highly accomplished men and women are admitted into their noble orders for people of title such as Ndi Ozo or Ndi Nze. These people receive insignia to show their stature. Membership is highly exclusive, and to qualify an individual need to be highly regarded and well-spoken of in the community.

=== Kola nut (Ọjị) ===
[[File:Kola nut.jpg|alt=Kola nut|thumb|Kola nut]]
Kola nut  occupies a unique position in the cultural life of Igbo people. Ọjị is the first thing served to any visitor in an Igbo home. Ọjị is served before an important function begins, be it marriage ceremony, settlement of family disputes or entering into any type of agreement. Ọjị is traditionally broken into pieces by hand, and if the Kola nut breaks into 3 pieces a special celebration is arranged.


== Module 4: Material Material Culture, Architecture, and Systems of Time ==

=== Igbo Architecture ===
Igbo architecture refers to the architectural styles and building traditions of the Igbo people. The architectural style is closely tied to the Igbo society's culture, beliefs, and social structure. While the architectural style has evolved, traditional Igbo architecture shares some common characteristics such as:

'''Compound layout'''- Igbo architectural traditions often revolve around the concept of a compound which is characterized by an enclosed area encompassing multiple family residences, open central courtyards, verandas, and auxiliary structures. These compounds are meticulously planned and sometimes paved with flat stones to foster communal living and facilitate familial engagements. Additionally, certain compounds feature unique elements like Impluvium houses, Gardens, Moats, and water wells demonstrating the diversity within Igbo architectural practices.

'''Ventilation''' - Igbo architecture integrates strategic placement of openings in buildings to promote cross-ventilation, aiding in regulating indoor temperatures. Employing expansive openings facilitates air circulation, ensuring occupant comfort. Depending on the area with high temperatures and humidity, evaporation of sweat becomes challenging; however, airflow aids this process, enhancing comfort. Moreover, construction practices involve thick walls, thatched roofs, and raised foundations to mitigate environmental challenges. The thick walls maintain cooler interiors in hot weather and warmth during rainy seasons. Thatched roofs provide insulation from direct sunlight, offering shade and contributing to thermal comfort.

'''Shrines and Sacred Spaces'''- Igbo architecture often includes designated spaces in compounds or community areas for ancestral shrines/temples and secret society meeting houses. These spaces are considered sacred and are an essential part of Igbo cultural and religious practices. These sacred structures may vary in design, ranging from simple open-air spaces to more elaborate structures with specific architectural features.

'''Decorative Elements -''' Traditional Igbo architecture often incorporates decorative elements, including painted designs on walls such as [[Uli (design)|uli]], carved wooden door frames, and intricate patterns on ceilings. These decorations may have symbolic or religious significance.

=== Traditional attire ===
Igbo traditional attire varies across regions of Southeastern and south south Nigeria with various cultural significance. 

'''&lt;big&gt;Men&lt;/big&gt;'''

For men, common garments include ''uwe mwuda'' or ''afe ntutu'' ( robe) or ''efe elu'', a basic shirt paired with underneath wrappers or skirts complemented by the ''okpu ozo'' (the feathered red cap), or ''Okpu aji'' (woolen cap), ''ofo'', ''mkpara'' (staff) and ''Akupe'' (handfans) for ceremonial or titled occasions while  loin clothes or waist wrappers were usually worn as casual wears or basic activities like hunting or farming. 

&lt;big&gt;'''Women'''&lt;/big&gt;

[[File:Igbo_woman_wearing_Akwete_obiakwa(double_wrapper)_with_uweobi(blouse)_and_Ichafu_headdress.jpg|thumb|Igbo women’s traditional attire showing Obiakwa(matching double wrappers) made of Akwete George, uweobi (blouse with puffed sleeves) and stiff Ichafu (headdress)]]

Traditional Igbo women's attire comprises many regional and age-based (''Ụmụagbọ'') variant, including the Obiakwa pair of matching wrappers, Uweobi (blouse), and Ịchafụ̀ (head-tie), an elaborate and voluminous headdress traditionally worn by mature women. 

[[File:Igbo_woman_wearing_Isiagu_obiakwa_maiden_attire,_aka_olu(coral_beads)_ngala(head_beads)_and_nza(horsetail).jpg|thumb|A short Obiakwa (wrapper-style) ensemble paired with a fitted blouse, complemented by nza (horsewhisk), ngala (head beads), and other beaded accessories. The textile features  Isiagu motif]]

Younger women may wear shorter Obiakwa wrappers paired with a tubular Uweobi blouse. Traditional adornments include ''aka olu'' (coral beads), ''ngala'' (head beads), and ''mgbaji'' (waist beads), often complemented by other ceremonial accessories like the ''akupe'' (hand fans) and ''nza'' (horsetail whisks). Traditional attire and adornment form many parts of Igbo cultural expressions associated with age, status, ceremony, and identity.

'''&lt;u&gt;Obiakwa&lt;/u&gt;''' 

The ''Obiakwa'' is a traditional women's double- wrapper attire unique to Igbo weaving traditions such as Akwete cloth. It consists of a pair of matching wrappers Descriptions of Akwete weaving note that such wrapper sets were engineered during the weaving process to be worn together.  It is therefore sold in matching pairs.  These wrappers are standardly paired with a blouse called uweobi and the Ichafu headdress. Younger women usually wear shorter ''obiakwa'' waist wrapper sets combined with fitted or tubular blouses or wrappers . These clothings are also complemented with ''ngala'', ''mbaji'' and ''aka'' (beaded accessories) as well ''uli'' body arts. In some regions, The Uli body art was also used to decorate both men and women in the form of lines forming patterns and shapes on the body. 

'''&lt;u&gt;Blouse&lt;/u&gt;''' 

To complete the silhouette, double wrappers(obiakwa) are paired with a Blouse (or ''uweobi''), a traditional fitted blouse. Short ''obiakwa'' styles are usually paired with tubular blouses. 


'''&lt;u&gt;Ichafu&lt;/u&gt;'''

[[File:Igbo_woman_wearing_Joojii_and_Ichafu._Igbo_regality.jpg|thumb|An Igbo woman dressed in traditional attire consisting of a white puff-sleeve Uweobi blouse, a red and gold double George wrapper, and a stiff, elaborately structured Ichafu headdress, accessorized with pearl jewelry.]]

''Ichafu'' is an elaborate head-tie or headdress worn by Igbo women, especially for church services, ceremonies and other social occasions.  It forms part of a broader clothing ensemble that may include wrappers, blouses and jewellery, and is typically tied in volumnious elevated layered styles with large folds and pleats rising above the head. Ichafu is tied with various textiles including synthetic damask, brocade, Akwete and George fabric, which gives it the stiff and highly elaborated look. In Ogadinma, published by Granta, women were described wearing colourful blouses with “expensive ichafu” tied “in layers and pleats until the scarves were piled atop their heads like large plants”. Other dialectical variations for Ichafu is ''Akwaisi'', ''ulari'', ''unari'', ''nsu n'isi'', ''ufu isi'', ''asusu isi'', ''nchafu isi,'' ''Akishi''.   

'''&lt;u&gt;Textiles&lt;/u&gt;'''

Textiles commonly used across Igbo land include ''Isiagu'' (often patterned with the tiger or Lion head motifs), ''Akwete'' and ''Akwaocha'' handwoven clothes, and richly patterned George wrappers.

[[File:Little world, Aichi prefecture - African plaza - Hat of a vassal - Ìgbo people in Nigeria - Collected in 2006.jpg|110px|thumb|left|A traditional Igbo hat made entirely from [[wool]].]]Women carried their babies on their backs with a strip of clothing binding the two with a knot at her chest. This baby carrying technique was and still is practiced by many people groups across Africa, including the Igbo. This method has been modernized in the form of the child carrier.  Both men and women wore wrappers.[[File:Igba nkwu ceremony 07.jpg|thumb|Igba nkwu, Igbo traditional marriage]]

[[File:Igbo Traditional marriage.jpg|thumb|Igbo Traditional Marriage attire]]

=== Calendar (Iguafo Igbo) ===
In the traditional Igbo calendar, a week has 4 days  (''Eke'', ''Orie'', ''Afọ'', ''Nkwọ''), seven weeks make one month, a month has 28 days and there are 13 months in a year. In the last month, an extra day is added. The names of the days have their roots in the mythology of the Kingdom of Nri. It was believed that Eri, the sky-born founder of the Nri kingdom, had gone on a journey to discover the mystery of time. On his journey he had saluted and counted the four days by the names of the spirits that governed them, and so the names of the spirits (''eke'', ''orie'', ''afọ'' and ''Nkwo'') became the days of the week.

{{col-begin}}{{col-2}}
{| class=&quot;wikitable&quot;
!No. || Months (Ọnwa) || Gregorian equivalent
|-
|1 || '''Ọnwa Mbụ''' || (3rd week of February)
|-
|2 || '''Ọnwa Abụa''' || (March)
|-
|3 || '''Ọnwa Ife Eke''' || (April)
|-
|4 || '''Ọnwa Anọ''' || (May)
|-
|5 || '''Ọnwa Agwụ''' || (June)
|-
|6 || '''Ọnwa Ifejiọkụ''' || (July)
|-
|7 || '''Ọnwa Alọm Chi''' || (August to early September)
|-
|8 || '''Ọnwa Ilo Mmụọ''' || (Late September)
|-
|9 || '''Ọnwa Ana''' || (October)
|-
|10 || '''Ọnwa Okike''' || (Early November)
|-
|11 || '''Ọnwa Ajana''' || (Late November)
|-
|12 || '''Ọnwa Ede Ajana''' || (Late November to December)
|-
|13 || '''Ọnwa Ụzọ Alụsị''' || (January to early February)&lt;ref&gt;{{cite book |last1=Onwuejeogwu |first1=M. Angulu |title=An Igbo Civilization: Nri Kingdom &amp; Hegemony |date=1981 |publisher=Ethnographica |isbn=978-978-123-105-6 }}{{page needed|date=January 2024}}&lt;/ref&gt;&lt;ref&gt;{{cite web |url=http://www.free-press-release.com/news/200802/1204305180.html |title=Eze Nri - Igu-Aro Festival - 1008th AD |publisher=Free-Press-Release Inc. |date=February 29, 2008 |access-date=2010-04-06 |archive-date=2009-07-24 |archive-url=https://web.archive.org/web/20090724025818/http://www.free-press-release.com/news/200802/1204305180.html |url-status=dead }}&lt;/ref&gt;
|}

{{col-break}}
An example of a month: '''''Ọnwa Mbụ'''''

{| class=&quot;wikitable&quot;
|-
!Eke || Orie || Afọ || Nkwọ
|-
||||| 1 || 2
|-
|3 || 4 || 5 || 6
|-
|7 || 8 || 9 || 10
|-
|11 || 12 || 13 || 14
|-
|15 || 16 || 17 || 18
|-
|19 || 20 || 21 || 22
|-
|23 || 24 || 25 || 26
|-
|27 || 28||||
|}
{{col-end}}

==== Naming after market days ====
Newborn babies were sometimes named after the day of the week when born. This is no longer the fashion.  Names such as ''Mgbeke'' (maiden [born] on the day of Eke), Mgborie (maiden [born] on the Orie day) are commonly seen among the Igbo people. For males, ''Mgbe'' is replaced by ''Nwa'' or &quot;Okoro&quot; (Igbo: Child [of]). Examples of this are Solomon Okoronkwo and Nwankwo Kanu, two popular footballers.


== Review and Independent Study Activities ==
The following prompts are provided for self-assessment and group discussion based entirely on the texts provided in the modules above:
1. List three traditional musical instruments utilized by the Igbo people and describe the stylistic origins of modern Igbo highlife music.
2. Outline the traditional cosmic structural divisions of the ancient Igbo religion (Odinani) and identify the primary role of an Alusi.
3. Detail the specific steps and cultural expectations tied to the realization of an &quot;Igbankwu Alumdi&quot; ceremony.
4. Synthesize the historical and modern developments of the &quot;Imu Ahia&quot; or &quot;Igba Boy&quot; system within the economic fabric of Igboland.

[[Category:Igbo culture]]
[[Category:Cultural Anthropology]]
[[Category:Learning Projects]]</text>
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      <timestamp>2026-06-07T06:20:09Z</timestamp>
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        <username>Wmbata</username>
        <id>3084293</id>
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      <text bytes="22366" sha1="o2vrw214gbl48snnopst6g7q5fmby3b" xml:space="preserve">== Introduction ==
'''Igbo culture''' are the customs, practices and traditions of the Igbo people of southeastern Nigeria. It consists of ancient practices known as ''Odinala'' ''ndi'' ''igbo'' as well as new concepts added into the Igbo culture either by cultural evolution or by outside influence. These customs and traditions includes the Igbo people's visual art, music and dance forms, as well as their attire, Food, cuisine and language dialects. Because of their various subgroups, the variety of their culture is heightened further.

== Learning Objectives ==
By reviewing this material, you should be able to:

* '''Identify the major traditional musical instruments and art forms of Ndị Igbo.'''
* '''Contrast the components of traditional Ndị Igbo cosmology.'''
* '''Detail the socio-economic functions of historical practices like traditional marriage, architecture, and the apprenticeship system.'''


== Module 1: Creative Arts and Expressive Traditions ==

=== Music ===
[[File:Udu.jpg|thumb|right|95px|Udu, an Igbo instrument]]
The Igbo people have a melodic and symphonic musical style. Instruments include Ọ̀pì  otherwise known as '''Oja''' a wind instrument similar to the flute, '''igba''', and '''ichaka'''.

Another popular musical form among Igbo people is highlife, which is a fusion of jazz and traditional music and widely popular in West Africa. The modern Igbo highlife is seen in the works of Prince Nico Mbarga, Dr Sir Warrior, Oliver De Coque, Bright Chimezie, Celestine Ukwu,Chief Osita Osadebe, And many others who are some of the greatest Igbo highlife musicians of the twentieth century. There are also other notable Igbo highlife artists, like the Mike Ejeagha, Paulson Kalu, Ali Chukwuma, Ozoemena Nwa Nsugbe.

=== Art ===
Igbo art is known for various types of masquerades, masks, outfits (symbolizing people), animals and abstract conceptions. Igbo art is also known for its bronze castings found in the town of Igbo Ukwu from the 9th century.

&lt;gallery widths=&quot;200&quot; heights=&quot;200&quot; mode=&quot;packed&quot;&gt;
File:Nigeria, igbo, maschera-elmo della società mmuo, xx secolo.jpg|Helmet-mask; 20th century; Indianapolis Museum of Art (USA)
File:Nigeria, igbo, figura femminile per un tempietto, xx secolo.jpg|Female figure for a small temple, 20th century; Indianapolis Museum of Art
File:Igbo brass anklet.jpg|Anklet beaten from a solid brass bar of the type worn by Igbo women. Now in the collection of Wolverhampton Art Gallery. The leg-tube extends approximately 7&amp;nbsp;cm each side of the 35&amp;nbsp;cm disc.
File:Bronze ceremonial vessel in form of a snail shell, 9th century, Igbo-Ukwu, Nigeria.JPG|Bronze ceremonial vessel in form of a snail shell; 9th century; from Igbo-Ukwu; Nigerian National Museum (Lagos, Nigeria)
File:Eze Onyiudo (2).jpg|Eze Onyiudo Masquerade Awka-Etiti
&lt;/gallery&gt;

=== Igbo masks and masquerades ===
There are two basic types of masquerades, visible and invisible. The visible masquerades are meant for the public. They often are more entertaining. Masks used offer a visual appeal for their shapes and forms. In these visible masquerades, performances of harassment, music, dance, and parodies are acted out (Oyeneke 25).

The invisible masquerades take place at night.  Sound is the main tool for them. The masquerader uses his voice to scream so it may be heard throughout the village. The masks used are usually fierce looking and their interpretation is only fully understood by the society's members.  These invisible masquerades call upon a silent village to strike fear in the hearts of those not initiated into their society.


== Module 2: Spiritual Beliefs and Cosmological Frameworks ==

=== Mythology ===
While today many Igbo people are Christian, the traditional ancient Igbo religion is known as Odinani. In the Igbo mythology, which is part of their ancient religion, the supreme God is called Chineke (&quot;the God of creation&quot;); Chineke created the world and everything in it and is associated with all things on Earth. To the ancient Igbo, the cosmos is divided into four complex parts:
 
* OKIKE (Creation)
* ALUSI (Supernatural Forces or Deities)
* MMUO (Spirit)
* UWA (World)

==== Alusi ====
[[File:Complex sculpture Nigeria BM Af1954 23 522 img02.jpg|thumb|alt=A photo of a complex wooden carving of animals, people and spirits laid on each other to about 2 meters in height|Complex wooden carving depicting images of power and daily life, such as horsemen, imported goods, military insignia, Europeans, rifles, wild beasts and masqueraders.]]'''Alusi''', also known as '''Arusi''' or '''Arushi''', are minor deities that are worshiped and served in Igbo mythology. There are a list of many different Alusi that exists within each community and each has its own purpose. When there is no longer need for the deity, it is returned to its source, through the help of a Chief Priest or Dibia, who is aware of the procedure and ensures that its done properly.

==== Mmuo ====
Mmuo simply means spirit. It is either a good and godly spirit (mmuo oma) or it is an evil spirit (mmuo ojo). For example, the Ogbanje spirit is seen as an evil spirit (mmuo ojo) and anyone possessed by this spirit is given spiritual attention. (Spiritual attention means a way of casting out the evil spirit through deliverance (Christian way) or through African Traditional Religion&amp;nbsp; (i.e. digging out his/her '''“iyi uwa”'''. the ATR way)). Ogbanje is an Igbo (Nigeria) term that means a repeater or someone who comes and departs.  Ogbanje is not a bad spirit in Igbo Cosmology. It is a word widely used to describe a kid or teenager who is claimed to die and be born repeatedly by the same person.

=== Osu caste system ===
Osu are a group of people whose ancestors were dedicated to serving in shrines and temples for the deities of the Igbo, and therefore were deemed property of the gods. Relationships and sometimes interactions with Osu were (and to this day, still are) in many cases, forbidden.

To this day being called an ''Osu'' remains a stigma that prevents people's progress and lifestyles.


== Module 3: Social Milestones and Economic Structures ==

=== Yam ===
The yam is very important to the Igbo as it is their staple crop. There are celebrations such as the New yam festival  which are held Every August of Every year for the harvesting of the yam.

The New Yam festival  is celebrated annually to secure a good harvest of the staple crop.  The festival is practiced primarily in Nigeria and other countries in West Africa.

=== Traditional marriage ===
Marriages in Igbo community follow a multi-step process before the bride and groom are proclaimed husband and wife in accordance with local law and tradition.
[[File:Igba nkwu ceremony 04.jpg|alt=Igbo Traditional Marriage|thumb|Traditional Igbo Marriage Attire]]
The traditional marriage is known as &quot;Igbankwu Alumdi&quot; in Igbo land, or wine carrying, since it involves the bride serving up a cup of palm wine to her fiancé. Prior to the wedding, the groom must go to the bride's compound with his father before the Igbankwu day to get the bride's father's consent to marry his daughter. If the bride's father is late, in this case, the bride's brother, uncle or male relative fills in for the bride's late father, as applies to the groom. On the second visit, when kola nuts (oji Igbo) are offered, the two fathers must arrange a price for the bride. In most cases, the bride's price is just symbolic, in addition to other requirements like kola nuts, goats, wine, fowl and so on. Normally, it takes more than one evening until the bride price is agreed upon, after which a feast is served to both parents. When the bride price is paid, another evening is set aside for the ceremony.

During the ceremony, the bride's father fills a cup with palm wine and hands it over to the daughter. Accompanied by her brides maids known as umuagbo nwunye, she then searches for the groom among the crowd of wedding guests to offer him the drink. Once the drink is offered, the bride and groom dance to the bride's father. They kneel before him and he will give them his blessings. After that, the couple dances for a while before taking their seats, then refreshment takes place followed by presentation of gifts, at times a speech from the MC, and then closing prayer and departure.

=== Apprenticeship ===
The Igbo have a unique form of apprenticeship in which either a male family member or a community member will spend time (usually in their teens to their adulthood) with another family, when they work for them. After the time spent with the family, the head of the host household, who is usually the older man who brought the apprentice into his household, will establish  the apprentice by either setting up a business for him or giving money or tools by which to make a living.

This practice was exploited by Europeans, who used this practice as a way of trading in enslaved people. Olaudah Equiano, although stolen from his home, was an Igbo person who was forced into service to an African family.  He said that he felt part of the family, unlike later, when he was shipped to North America and enslaved in the Thirteen Colonies.

The Igbo apprenticeship system is called Imu Ahia or Igba Boy in Igboland. It became more prominent among the Igbos after the Nigerian civil war, in a quest to survive the £20 policy which was proposed by Obafemi Awolowo that only £20 be given to every Biafran citizen to survive on regardless of what they had in the bank before the war and the rest of the money were held by the Nigerian government.

Petty trade was one of the only ways to build back destroyed communities as well as farming, but then, farming required time that was not readily available at that moment. Essentially, most people went into trading.

This Imu-Ahia/Igba Boy model was simple, it works in such a way that business owners would take in younger boys which can be relative, sibling or non-relative from same region, house them and have them work as apprentices in business while learning how it works and the secrets of the business. After the allotted time for the training was reached, 5–8 years’ time, a little graduation ceremony would be held for the '''Nwa Boy''' (the person that learnt the trade). He would also be paid a lump sum for their services over the years, and the money will be used to start a business for the '''Nwa Boy'''.

=== Chieftaincy Title ===
[[File:Igbo ichi marks.jpg|thumb|An Igbo man with ''Ichi'' marks, a sign of rank as an Ozo]]
Highly accomplished men and women are admitted into their noble orders for people of title such as Ndi Ozo or Ndi Nze. These people receive insignia to show their stature. Membership is highly exclusive, and to qualify an individual need to be highly regarded and well-spoken of in the community.

=== Kola nut (Ọjị) ===
[[File:Kola nut.jpg|alt=Kola nut|thumb|Kola nut]]
Kola nut  occupies a unique position in the cultural life of Igbo people. Ọjị is the first thing served to any visitor in an Igbo home. Ọjị is served before an important function begins, be it marriage ceremony, settlement of family disputes or entering into any type of agreement. Ọjị is traditionally broken into pieces by hand, and if the Kola nut breaks into 3 pieces a special celebration is arranged.


== Module 4: Material, Material Culture, Architecture, and Systems of Time ==

=== Igbo Architecture ===
Igbo architecture refers to the architectural styles and building traditions of the Igbo people. The architectural style is closely tied to the Igbo society's culture, beliefs, and social structure. While the architectural style has evolved, traditional Igbo architecture shares some common characteristics such as:

'''Compound layout'''- Igbo architectural traditions often revolve around the concept of a compound which is characterized by an enclosed area encompassing multiple family residences, open central courtyards, verandas, and auxiliary structures. These compounds are meticulously planned and sometimes paved with flat stones to foster communal living and facilitate familial engagements. Additionally, certain compounds feature unique elements like Impluvium houses, Gardens, Moats, and water wells demonstrating the diversity within Igbo architectural practices.

'''Ventilation''' - Igbo architecture integrates strategic placement of openings in buildings to promote cross-ventilation, aiding in regulating indoor temperatures. Employing expansive openings facilitates air circulation, ensuring occupant comfort. Depending on the area with high temperatures and humidity, evaporation of sweat becomes challenging; however, airflow aids this process, enhancing comfort. Moreover, construction practices involve thick walls, thatched roofs, and raised foundations to mitigate environmental challenges. The thick walls maintain cooler interiors in hot weather and warmth during rainy seasons. Thatched roofs provide insulation from direct sunlight, offering shade and contributing to thermal comfort.

'''Shrines and Sacred Spaces'''- Igbo architecture often includes designated spaces in compounds or community areas for ancestral shrines/temples and secret society meeting houses. These spaces are considered sacred and are an essential part of Igbo cultural and religious practices. These sacred structures may vary in design, ranging from simple open-air spaces to more elaborate structures with specific architectural features.

'''Decorative Elements -''' Traditional Igbo architecture often incorporates decorative elements, including painted designs on walls such as [[Uli (design)|uli]], carved wooden door frames, and intricate patterns on ceilings. These decorations may have symbolic or religious significance.

=== Traditional attire ===
Igbo traditional attire varies across regions of Southeastern and south south Nigeria with various cultural significance. 

'''&lt;big&gt;Men&lt;/big&gt;'''

For men, common garments include ''uwe mwuda'' or ''afe ntutu'' ( robe) or ''efe elu'', a basic shirt paired with underneath wrappers or skirts complemented by the ''okpu ozo'' (the feathered red cap), or ''Okpu aji'' (woolen cap), ''ofo'', ''mkpara'' (staff) and ''Akupe'' (handfans) for ceremonial or titled occasions while  loin clothes or waist wrappers were usually worn as casual wears or basic activities like hunting or farming. 

&lt;big&gt;'''Women'''&lt;/big&gt;

[[File:Igbo_woman_wearing_Akwete_obiakwa(double_wrapper)_with_uweobi(blouse)_and_Ichafu_headdress.jpg|thumb|Igbo women’s traditional attire showing Obiakwa(matching double wrappers) made of Akwete George, uweobi (blouse with puffed sleeves) and stiff Ichafu (headdress)]]

Traditional Igbo women's attire comprises many regional and age-based (''Ụmụagbọ'') variant, including the Obiakwa pair of matching wrappers, Uweobi (blouse), and Ịchafụ̀ (head-tie), an elaborate and voluminous headdress traditionally worn by mature women. 

[[File:Igbo_woman_wearing_Isiagu_obiakwa_maiden_attire,_aka_olu(coral_beads)_ngala(head_beads)_and_nza(horsetail).jpg|thumb|A short Obiakwa (wrapper-style) ensemble paired with a fitted blouse, complemented by nza (horsewhisk), ngala (head beads), and other beaded accessories. The textile features  Isiagu motif]]

Younger women may wear shorter Obiakwa wrappers paired with a tubular Uweobi blouse. Traditional adornments include ''aka olu'' (coral beads), ''ngala'' (head beads), and ''mgbaji'' (waist beads), often complemented by other ceremonial accessories like the ''akupe'' (hand fans) and ''nza'' (horsetail whisks). Traditional attire and adornment form many parts of Igbo cultural expressions associated with age, status, ceremony, and identity.

'''&lt;u&gt;Obiakwa&lt;/u&gt;''' 

The ''Obiakwa'' is a traditional women's double- wrapper attire unique to Igbo weaving traditions such as Akwete cloth. It consists of a pair of matching wrappers Descriptions of Akwete weaving note that such wrapper sets were engineered during the weaving process to be worn together.  It is therefore sold in matching pairs.  These wrappers are standardly paired with a blouse called uweobi and the Ichafu headdress. Younger women usually wear shorter ''obiakwa'' waist wrapper sets combined with fitted or tubular blouses or wrappers . These clothings are also complemented with ''ngala'', ''mbaji'' and ''aka'' (beaded accessories) as well ''uli'' body arts. In some regions, The Uli body art was also used to decorate both men and women in the form of lines forming patterns and shapes on the body. 

'''&lt;u&gt;Blouse&lt;/u&gt;''' 

To complete the silhouette, double wrappers(obiakwa) are paired with a Blouse (or ''uweobi''), a traditional fitted blouse. Short ''obiakwa'' styles are usually paired with tubular blouses. 


'''&lt;u&gt;Ichafu&lt;/u&gt;'''

[[File:Igbo_woman_wearing_Joojii_and_Ichafu._Igbo_regality.jpg|thumb|An Igbo woman dressed in traditional attire consisting of a white puff-sleeve Uweobi blouse, a red and gold double George wrapper, and a stiff, elaborately structured Ichafu headdress, accessorized with pearl jewelry.]]

''Ichafu'' is an elaborate head-tie or headdress worn by Igbo women, especially for church services, ceremonies and other social occasions.  It forms part of a broader clothing ensemble that may include wrappers, blouses and jewellery, and is typically tied in volumnious elevated layered styles with large folds and pleats rising above the head. Ichafu is tied with various textiles including synthetic damask, brocade, Akwete and George fabric, which gives it the stiff and highly elaborated look. In Ogadinma, published by Granta, women were described wearing colourful blouses with “expensive ichafu” tied “in layers and pleats until the scarves were piled atop their heads like large plants”. Other dialectical variations for Ichafu is ''Akwaisi'', ''ulari'', ''unari'', ''nsu n'isi'', ''ufu isi'', ''asusu isi'', ''nchafu isi,'' ''Akishi''.   

'''&lt;u&gt;Textiles&lt;/u&gt;'''

Textiles commonly used across Igbo land include ''Isiagu'' (often patterned with the tiger or Lion head motifs), ''Akwete'' and ''Akwaocha'' handwoven clothes, and richly patterned George wrappers.

[[File:Little world, Aichi prefecture - African plaza - Hat of a vassal - Ìgbo people in Nigeria - Collected in 2006.jpg|110px|thumb|left|A traditional Igbo hat made entirely from [[wool]].]]Women carried their babies on their backs with a strip of clothing binding the two with a knot at her chest. This baby carrying technique was and still is practiced by many people groups across Africa, including the Igbo. This method has been modernized in the form of the child carrier.  Both men and women wore wrappers.[[File:Igba nkwu ceremony 07.jpg|thumb|Igba nkwu, Igbo traditional marriage]]

[[File:Igbo Traditional marriage.jpg|thumb|Igbo Traditional Marriage attire]]

=== Calendar (Iguafo Igbo) ===
In the traditional Igbo calendar, a week has 4 days  (''Eke'', ''Orie'', ''Afọ'', ''Nkwọ''), seven weeks make one month, a month has 28 days and there are 13 months in a year. In the last month, an extra day is added. The names of the days have their roots in the mythology of the Kingdom of Nri. It was believed that Eri, the sky-born founder of the Nri kingdom, had gone on a journey to discover the mystery of time. On his journey he had saluted and counted the four days by the names of the spirits that governed them, and so the names of the spirits (''eke'', ''orie'', ''afọ'' and ''Nkwo'') became the days of the week.

{{col-begin}}{{col-2}}
{| class=&quot;wikitable&quot;
!No. || Months (Ọnwa) || Gregorian equivalent
|-
|1 || '''Ọnwa Mbụ''' || (3rd week of February)
|-
|2 || '''Ọnwa Abụa''' || (March)
|-
|3 || '''Ọnwa Ife Eke''' || (April)
|-
|4 || '''Ọnwa Anọ''' || (May)
|-
|5 || '''Ọnwa Agwụ''' || (June)
|-
|6 || '''Ọnwa Ifejiọkụ''' || (July)
|-
|7 || '''Ọnwa Alọm Chi''' || (August to early September)
|-
|8 || '''Ọnwa Ilo Mmụọ''' || (Late September)
|-
|9 || '''Ọnwa Ana''' || (October)
|-
|10 || '''Ọnwa Okike''' || (Early November)
|-
|11 || '''Ọnwa Ajana''' || (Late November)
|-
|12 || '''Ọnwa Ede Ajana''' || (Late November to December)
|-
|13 || '''Ọnwa Ụzọ Alụsị''' || (January to early February)&lt;ref&gt;{{cite book |last1=Onwuejeogwu |first1=M. Angulu |title=An Igbo Civilization: Nri Kingdom &amp; Hegemony |date=1981 |publisher=Ethnographica |isbn=978-978-123-105-6 }}{{page needed|date=January 2024}}&lt;/ref&gt;&lt;ref&gt;{{cite web |url=http://www.free-press-release.com/news/200802/1204305180.html |title=Eze Nri - Igu-Aro Festival - 1008th AD |publisher=Free-Press-Release Inc. |date=February 29, 2008 |access-date=2010-04-06 |archive-date=2009-07-24 |archive-url=https://web.archive.org/web/20090724025818/http://www.free-press-release.com/news/200802/1204305180.html |url-status=dead }}&lt;/ref&gt;
|}

{{col-break}}
An example of a month: '''''Ọnwa Mbụ'''''

{| class=&quot;wikitable&quot;
|-
!Eke || Orie || Afọ || Nkwọ
|-
||||| 1 || 2
|-
|3 || 4 || 5 || 6
|-
|7 || 8 || 9 || 10
|-
|11 || 12 || 13 || 14
|-
|15 || 16 || 17 || 18
|-
|19 || 20 || 21 || 22
|-
|23 || 24 || 25 || 26
|-
|27 || 28||||
|}
{{col-end}}

==== Naming after market days ====
Newborn babies were sometimes named after the day of the week when born. This is no longer the fashion.  Names such as ''Mgbeke'' (maiden [born] on the day of Eke), Mgborie (maiden [born] on the Orie day) are commonly seen among the Igbo people. For males, ''Mgbe'' is replaced by ''Nwa'' or &quot;Okoro&quot; (Igbo: Child [of]). Examples of this are Solomon Okoronkwo and Nwankwo Kanu, two popular footballers.


== Review and Independent Study Activities ==
The following prompts are provided for self-assessment and group discussion based entirely on the texts provided in the modules above:
1. List three traditional musical instruments utilized by the Igbo people and describe the stylistic origins of modern Igbo highlife music.
2. Outline the traditional cosmic structural divisions of the ancient Igbo religion (Odinani) and identify the primary role of an Alusi.
3. Detail the specific steps and cultural expectations tied to the realization of an &quot;Igbankwu Alumdi&quot; ceremony.
4. Synthesize the historical and modern developments of the '''Imu Ahia''' or '''Igba Boy''' system within the economic fabric of Igboland.

[[Category:Igbo culture]]
[[Category:Cultural Anthropology]]
[[Category:Learning Projects]]</text>
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    <revision>
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      <timestamp>2026-06-07T06:28:12Z</timestamp>
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      <text bytes="21654" sha1="e08boaquqd434tkyw12mxsiv9ga1xlf" xml:space="preserve">== Introduction ==
'''Igbo culture''' are the customs, practices and traditions of the Igbo people of southeastern Nigeria. It consists of ancient practices known as ''Odinala'' ''ndi'' ''igbo'' as well as new concepts added into the Igbo culture either by cultural evolution or by outside influence. These customs and traditions includes the Igbo people's visual art, music and dance forms, as well as their attire, Food, cuisine and language dialects. Because of their various subgroups, the variety of their culture is heightened further.

== Learning Objectives ==
By reviewing this material, you should be able to:

* '''Identify the major traditional musical instruments and art forms of Ndị Igbo.'''
* '''Contrast the components of traditional Ndị Igbo cosmology.'''
* '''Detail the socio-economic functions of historical practices like traditional marriage, architecture, and the apprenticeship system.'''


== Module 1: Creative Arts and Expressive Traditions ==

=== Music ===
[[File:Udu.jpg|thumb|right|95px|Udu, an Igbo instrument]]
The Igbo people have a melodic and symphonic musical style. Instruments include Ọ̀pì  otherwise known as '''Oja''' a wind instrument similar to the flute, '''igba''', and '''ichaka'''.

Another popular musical form among Igbo people is highlife, which is a fusion of jazz and traditional music and widely popular in West Africa. The modern Igbo highlife is seen in the works of Prince Nico Mbarga, Dr Sir Warrior, Oliver De Coque, Bright Chimezie, Celestine Ukwu,Chief Osita Osadebe, And many others who are some of the greatest Igbo highlife musicians of the twentieth century. There are also other notable Igbo highlife artists, like the Mike Ejeagha, Paulson Kalu, Ali Chukwuma, Ozoemena Nwa Nsugbe.

=== Art ===
Igbo art is known for various types of masquerades, masks, outfits (symbolizing people), animals and abstract conceptions. Igbo art is also known for its bronze castings found in the town of Igbo Ukwu from the 9th century.

&lt;gallery widths=&quot;200&quot; heights=&quot;200&quot; mode=&quot;packed&quot;&gt;
File:Nigeria, igbo, maschera-elmo della società mmuo, xx secolo.jpg|Helmet-mask; 20th century; Indianapolis Museum of Art (USA)
File:Nigeria, igbo, figura femminile per un tempietto, xx secolo.jpg|Female figure for a small temple, 20th century; Indianapolis Museum of Art
File:Igbo brass anklet.jpg|Anklet beaten from a solid brass bar of the type worn by Igbo women. Now in the collection of Wolverhampton Art Gallery. The leg-tube extends approximately 7&amp;nbsp;cm each side of the 35&amp;nbsp;cm disc.
File:Bronze ceremonial vessel in form of a snail shell, 9th century, Igbo-Ukwu, Nigeria.JPG|Bronze ceremonial vessel in form of a snail shell; 9th century; from Igbo-Ukwu; Nigerian National Museum (Lagos, Nigeria)
File:Eze Onyiudo (2).jpg|Eze Onyiudo Masquerade Awka-Etiti
&lt;/gallery&gt;

=== Igbo masks and masquerades ===
There are two basic types of masquerades, visible and invisible. The visible masquerades are meant for the public. They often are more entertaining. Masks used offer a visual appeal for their shapes and forms. In these visible masquerades, performances of harassment, music, dance, and parodies are acted out (Oyeneke 25).

The invisible masquerades take place at night.  Sound is the main tool for them. The masquerader uses his voice to scream so it may be heard throughout the village. The masks used are usually fierce looking and their interpretation is only fully understood by the society's members.  These invisible masquerades call upon a silent village to strike fear in the hearts of those not initiated into their society.


== Module 2: Spiritual Beliefs and Cosmological Frameworks ==

=== Mythology ===
While today many Igbo people are Christian, the traditional ancient Igbo religion is known as Odinani. In the Igbo mythology, which is part of their ancient religion, the supreme God is called Chineke (&quot;the God of creation&quot;); Chineke created the world and everything in it and is associated with all things on Earth. To the ancient Igbo, the cosmos is divided into four complex parts:
 
* OKIKE (Creation)
* ALUSI (Supernatural Forces or Deities)
* MMUO (Spirit)
* UWA (World)

==== Alusi ====
[[File:Complex sculpture Nigeria BM Af1954 23 522 img02.jpg|thumb|alt=A photo of a complex wooden carving of animals, people and spirits laid on each other to about 2 meters in height|Complex wooden carving depicting images of power and daily life, such as horsemen, imported goods, military insignia, Europeans, rifles, wild beasts and masqueraders.]]'''Alusi''', also known as '''Arusi''' or '''Arushi''', are minor deities that are worshiped and served in Igbo mythology. There are a list of many different Alusi that exists within each community and each has its own purpose. When there is no longer need for the deity, it is returned to its source, through the help of a Chief Priest or Dibia, who is aware of the procedure and ensures that its done properly.

==== Mmuo ====
Mmuo simply means spirit. It is either a good and godly spirit (mmuo oma) or it is an evil spirit (mmuo ojo). For example, the Ogbanje spirit is seen as an evil spirit (mmuo ojo) and anyone possessed by this spirit is given spiritual attention. (Spiritual attention means a way of casting out the evil spirit through deliverance (Christian way) or through African Traditional Religion&amp;nbsp; (i.e. digging out his/her '''“iyi uwa”'''. the ATR way)). Ogbanje is an Igbo (Nigeria) term that means a repeater or someone who comes and departs.  Ogbanje is not a bad spirit in Igbo Cosmology. It is a word widely used to describe a kid or teenager who is claimed to die and be born repeatedly by the same person.

=== Osu caste system ===
Osu are a group of people whose ancestors were dedicated to serving in shrines and temples for the deities of the Igbo, and therefore were deemed property of the gods. Relationships and sometimes interactions with Osu were (and to this day, still are) in many cases, forbidden.

To this day being called an ''Osu'' remains a stigma that prevents people's progress and lifestyles.


== Module 3: Social Milestones and Economic Structures ==

=== Yam ===
The yam is very important to the Igbo as it is their staple crop. There are celebrations such as the New yam festival  which are held Every August of Every year for the harvesting of the yam.

The New Yam festival  is celebrated annually to secure a good harvest of the staple crop.  The festival is practiced primarily in Nigeria and other countries in West Africa.

=== Traditional marriage ===
Marriages in Igbo community follow a multi-step process before the bride and groom are proclaimed husband and wife in accordance with local law and tradition.
[[File:Igba nkwu ceremony 04.jpg|alt=Igbo Traditional Marriage|thumb|Traditional Igbo Marriage Attire]]
The traditional marriage is known as &quot;Igbankwu Alumdi&quot; in Igbo land, or wine carrying, since it involves the bride serving up a cup of palm wine to her fiancé. Prior to the wedding, the groom must go to the bride's compound with his father before the Igbankwu day to get the bride's father's consent to marry his daughter. If the bride's father is late, in this case, the bride's brother, uncle or male relative fills in for the bride's late father, as applies to the groom. On the second visit, when kola nuts (oji Igbo) are offered, the two fathers must arrange a price for the bride. In most cases, the bride's price is just symbolic, in addition to other requirements like kola nuts, goats, wine, fowl and so on. Normally, it takes more than one evening until the bride price is agreed upon, after which a feast is served to both parents. When the bride price is paid, another evening is set aside for the ceremony.

During the ceremony, the bride's father fills a cup with palm wine and hands it over to the daughter. Accompanied by her brides maids known as umuagbo nwunye, she then searches for the groom among the crowd of wedding guests to offer him the drink. Once the drink is offered, the bride and groom dance to the bride's father. They kneel before him and he will give them his blessings. After that, the couple dances for a while before taking their seats, then refreshment takes place followed by presentation of gifts, at times a speech from the MC, and then closing prayer and departure.

=== Apprenticeship ===
The Igbo have a unique form of apprenticeship in which either a male family member or a community member will spend time (usually in their teens to their adulthood) with another family, when they work for them. After the time spent with the family, the head of the host household, who is usually the older man who brought the apprentice into his household, will establish  the apprentice by either setting up a business for him or giving money or tools by which to make a living.

This practice was exploited by Europeans, who used this practice as a way of trading in enslaved people. Olaudah Equiano, although stolen from his home, was an Igbo person who was forced into service to an African family.  He said that he felt part of the family, unlike later, when he was shipped to North America and enslaved in the Thirteen Colonies.

The Igbo apprenticeship system is called Imu Ahia or Igba Boy in Igboland. It became more prominent among the Igbos after the Nigerian civil war, in a quest to survive the £20 policy which was proposed by Obafemi Awolowo that only £20 be given to every Biafran citizen to survive on regardless of what they had in the bank before the war and the rest of the money were held by the Nigerian government.

Petty trade was one of the only ways to build back destroyed communities as well as farming, but then, farming required time that was not readily available at that moment. Essentially, most people went into trading.

This Imu-Ahia/Igba Boy model was simple, it works in such a way that business owners would take in younger boys which can be relative, sibling or non-relative from same region, house them and have them work as apprentices in business while learning how it works and the secrets of the business. After the allotted time for the training was reached, 5–8 years’ time, a little graduation ceremony would be held for the '''Nwa Boy''' (the person that learnt the trade). He would also be paid a lump sum for their services over the years, and the money will be used to start a business for the '''Nwa Boy'''.

=== Chieftaincy Title ===
[[File:Igbo ichi marks.jpg|thumb|An Igbo man with ''Ichi'' marks, a sign of rank as an Ozo]]
Highly accomplished men and women are admitted into their noble orders for people of title such as Ndi Ozo or Ndi Nze. These people receive insignia to show their stature. Membership is highly exclusive, and to qualify an individual need to be highly regarded and well-spoken of in the community.

=== Kola nut (Ọjị) ===
[[File:Kola nut.jpg|alt=Kola nut|thumb|Kola nut]]
Kola nut  occupies a unique position in the cultural life of Igbo people. Ọjị is the first thing served to any visitor in an Igbo home. Ọjị is served before an important function begins, be it marriage ceremony, settlement of family disputes or entering into any type of agreement. Ọjị is traditionally broken into pieces by hand, and if the Kola nut breaks into 3 pieces a special celebration is arranged.


== Module 4: Material, Material Culture, Architecture, and Systems of Time ==

=== Igbo Architecture ===
Igbo architecture refers to the architectural styles and building traditions of the Igbo people. The architectural style is closely tied to the Igbo society's culture, beliefs, and social structure. While the architectural style has evolved, traditional Igbo architecture shares some common characteristics such as:

'''Compound layout'''- Igbo architectural traditions often revolve around the concept of a compound which is characterized by an enclosed area encompassing multiple family residences, open central courtyards, verandas, and auxiliary structures. These compounds are meticulously planned and sometimes paved with flat stones to foster communal living and facilitate familial engagements. Additionally, certain compounds feature unique elements like Impluvium houses, Gardens, Moats, and water wells demonstrating the diversity within Igbo architectural practices.

'''Ventilation''' - Igbo architecture integrates strategic placement of openings in buildings to promote cross-ventilation, aiding in regulating indoor temperatures. Employing expansive openings facilitates air circulation, ensuring occupant comfort. Depending on the area with high temperatures and humidity, evaporation of sweat becomes challenging; however, airflow aids this process, enhancing comfort. Moreover, construction practices involve thick walls, thatched roofs, and raised foundations to mitigate environmental challenges. The thick walls maintain cooler interiors in hot weather and warmth during rainy seasons. Thatched roofs provide insulation from direct sunlight, offering shade and contributing to thermal comfort.

'''Shrines and Sacred Spaces'''- Igbo architecture often includes designated spaces in compounds or community areas for ancestral shrines/temples and secret society meeting houses. These spaces are considered sacred and are an essential part of Igbo cultural and religious practices. These sacred structures may vary in design, ranging from simple open-air spaces to more elaborate structures with specific architectural features.

'''Decorative Elements -''' Traditional Igbo architecture often incorporates decorative elements, including painted designs on walls such as [[Uli (design)|uli]], carved wooden door frames, and intricate patterns on ceilings. These decorations may have symbolic or religious significance.

=== Traditional attire ===
Igbo traditional attire varies across regions of Southeastern and south south Nigeria with various cultural significance. 

'''&lt;big&gt;Men&lt;/big&gt;'''

For men, common garments include ''uwe mwuda'' or ''afe ntutu'' ( robe) or ''efe elu'', a basic shirt paired with underneath wrappers or skirts complemented by the ''okpu ozo'' (the feathered red cap), or ''Okpu aji'' (woolen cap), ''ofo'', ''mkpara'' (staff) and ''Akupe'' (handfans) for ceremonial or titled occasions while  loin clothes or waist wrappers were usually worn as casual wears or basic activities like hunting or farming. 

&lt;big&gt;'''Women'''&lt;/big&gt;

[[File:Igbo_woman_wearing_Akwete_obiakwa(double_wrapper)_with_uweobi(blouse)_and_Ichafu_headdress.jpg|thumb|Igbo women’s traditional attire showing Obiakwa(matching double wrappers) made of Akwete George, uweobi (blouse with puffed sleeves) and stiff Ichafu (headdress)]]

Traditional Igbo women's attire comprises many regional and age-based (''Ụmụagbọ'') variant, including the Obiakwa pair of matching wrappers, Uweobi (blouse), and Ịchafụ̀ (head-tie), an elaborate and voluminous headdress traditionally worn by mature women. 

[[File:Igbo_woman_wearing_Isiagu_obiakwa_maiden_attire,_aka_olu(coral_beads)_ngala(head_beads)_and_nza(horsetail).jpg|thumb|A short Obiakwa (wrapper-style) ensemble paired with a fitted blouse, complemented by nza (horsewhisk), ngala (head beads), and other beaded accessories. The textile features  Isiagu motif]]

Younger women may wear shorter Obiakwa wrappers paired with a tubular Uweobi blouse. Traditional adornments include ''aka olu'' (coral beads), ''ngala'' (head beads), and ''mgbaji'' (waist beads), often complemented by other ceremonial accessories like the ''akupe'' (hand fans) and ''nza'' (horsetail whisks). Traditional attire and adornment form many parts of Igbo cultural expressions associated with age, status, ceremony, and identity.

'''&lt;u&gt;Obiakwa&lt;/u&gt;''' 

The ''Obiakwa'' is a traditional women's double- wrapper attire unique to Igbo weaving traditions such as Akwete cloth. It consists of a pair of matching wrappers Descriptions of Akwete weaving note that such wrapper sets were engineered during the weaving process to be worn together.  It is therefore sold in matching pairs.  These wrappers are standardly paired with a blouse called uweobi and the Ichafu headdress. Younger women usually wear shorter ''obiakwa'' waist wrapper sets combined with fitted or tubular blouses or wrappers . These clothings are also complemented with ''ngala'', ''mbaji'' and ''aka'' (beaded accessories) as well ''uli'' body arts. In some regions, The Uli body art was also used to decorate both men and women in the form of lines forming patterns and shapes on the body. 

'''&lt;u&gt;Blouse&lt;/u&gt;''' 

To complete the silhouette, double wrappers(obiakwa) are paired with a Blouse (or ''uweobi''), a traditional fitted blouse. Short ''obiakwa'' styles are usually paired with tubular blouses. 


'''&lt;u&gt;Ichafu&lt;/u&gt;'''

[[File:Igbo_woman_wearing_Joojii_and_Ichafu._Igbo_regality.jpg|thumb|An Igbo woman dressed in traditional attire consisting of a white puff-sleeve Uweobi blouse, a red and gold double George wrapper, and a stiff, elaborately structured Ichafu headdress, accessorized with pearl jewelry.]]

''Ichafu'' is an elaborate head-tie or headdress worn by Igbo women, especially for church services, ceremonies and other social occasions.  It forms part of a broader clothing ensemble that may include wrappers, blouses and jewellery, and is typically tied in volumnious elevated layered styles with large folds and pleats rising above the head. Ichafu is tied with various textiles including synthetic damask, brocade, Akwete and George fabric, which gives it the stiff and highly elaborated look. In Ogadinma, published by Granta, women were described wearing colourful blouses with “expensive ichafu” tied “in layers and pleats until the scarves were piled atop their heads like large plants”. Other dialectical variations for Ichafu is ''Akwaisi'', ''ulari'', ''unari'', ''nsu n'isi'', ''ufu isi'', ''asusu isi'', ''nchafu isi,'' ''Akishi''.   

'''&lt;u&gt;Textiles&lt;/u&gt;'''

Textiles commonly used across Igbo land include ''Isiagu'' (often patterned with the tiger or Lion head motifs), ''Akwete'' and ''Akwaocha'' handwoven clothes, and richly patterned George wrappers.

[[File:Little world, Aichi prefecture - African plaza - Hat of a vassal - Ìgbo people in Nigeria - Collected in 2006.jpg|110px|thumb|left|A traditional Igbo hat made entirely from [[wool]].]]Women carried their babies on their backs with a strip of clothing binding the two with a knot at her chest. This baby carrying technique was and still is practiced by many people groups across Africa, including the Igbo. This method has been modernized in the form of the child carrier.  Both men and women wore wrappers.[[File:Igba nkwu ceremony 07.jpg|thumb|Igba nkwu, Igbo traditional marriage]]

[[File:Igbo Traditional marriage.jpg|thumb|Igbo Traditional Marriage attire]]

=== Calendar (Iguafo Igbo) ===
In the traditional Igbo calendar, a week has 4 days  (''Eke'', ''Orie'', ''Afọ'', ''Nkwọ''), seven weeks make one month, a month has 28 days and there are 13 months in a year. In the last month, an extra day is added. The names of the days have their roots in the mythology of the Kingdom of Nri. It was believed that Eri, the sky-born founder of the Nri kingdom, had gone on a journey to discover the mystery of time. On his journey he had saluted and counted the four days by the names of the spirits that governed them, and so the names of the spirits (''eke'', ''orie'', ''afọ'' and ''Nkwo'') became the days of the week.

{{col-begin}}{{col-2}}
{| class=&quot;wikitable&quot;
!No. || Months (Ọnwa) || Gregorian equivalent
|-
|1 || '''Ọnwa Mbụ''' || (3rd week of February)
|-
|2 || '''Ọnwa Abụa''' || (March)
|-
|3 || '''Ọnwa Ife Eke''' || (April)
|-
|4 || '''Ọnwa Anọ''' || (May)
|-
|5 || '''Ọnwa Agwụ''' || (June)
|-
|6 || '''Ọnwa Ifejiọkụ''' || (July)
|-
|7 || '''Ọnwa Alọm Chi''' || (August to early September)
|-
|8 || '''Ọnwa Ilo Mmụọ''' || (Late September)
|-
|9 || '''Ọnwa Ana''' || (October)
|-
|10 || '''Ọnwa Okike''' || (Early November)
|-
|11 || '''Ọnwa Ajana''' || (Late November)
|-
|12 || '''Ọnwa Ede Ajana''' || (Late November to December)
|-
|13 || '''Ọnwa Ụzọ Alụsị''' || (January to early February)&lt;ref&gt;{{cite book |last1=Onwuejeogwu |first1=M. Angulu |title=An Igbo Civilization: Nri Kingdom &amp; Hegemony |date=1981 |publisher=Ethnographica |isbn=978-978-123-105-6 }}{{page needed|date=January 2024}}&lt;/ref&gt;&lt;ref&gt;{{cite web |url=http://www.free-press-release.com/news/200802/1204305180.html |title=Eze Nri - Igu-Aro Festival - 1008th AD |publisher=Free-Press-Release Inc. |date=February 29, 2008 |access-date=2010-04-06 |archive-date=2009-07-24 |archive-url=https://web.archive.org/web/20090724025818/http://www.free-press-release.com/news/200802/1204305180.html |url-status=dead }}&lt;/ref&gt;
|}

{{col-break}}
An example of a month: '''''Ọnwa Mbụ'''''

{| class=&quot;wikitable&quot;
|-
!Eke || Orie || Afọ || Nkwọ
|-
||||| 1 || 2
|-
|3 || 4 || 5 || 6
|-
|7 || 8 || 9 || 10
|-
|11 || 12 || 13 || 14
|-
|15 || 16 || 17 || 18
|-
|19 || 20 || 21 || 22
|-
|23 || 24 || 25 || 26
|-
|27 || 28||||
|}
{{col-end}}

==== Naming after market days ====
Newborn babies were sometimes named after the day of the week when born. This is no longer the fashion.  Names such as ''Mgbeke'' (maiden [born] on the day of Eke), Mgborie (maiden [born] on the Orie day) are commonly seen among the Igbo people. For males, ''Mgbe'' is replaced by ''Nwa'' or &quot;Okoro&quot; (Igbo: Child [of]). Examples of this are Solomon Okoronkwo and Nwankwo Kanu, two popular footballers.


[[Category:Igbo culture]]
[[Category:Cultural Anthropology]]
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      <text bytes="21697" sha1="rp64q9qbrznrio858b50ka8sffolcxn" xml:space="preserve">== Introduction ==
'''Igbo culture''' are the customs, practices and traditions of the Igbo people of southeastern Nigeria. It consists of ancient practices known as ''Odinala'' ''ndi'' ''igbo'' as well as new concepts added into the Igbo culture either by cultural evolution or by outside influence. These customs and traditions includes the Igbo people's visual art, music and dance forms, as well as their attire, Food, cuisine and language dialects. Because of their various subgroups, the variety of their culture is heightened further.

== Learning Objectives ==
By reviewing this material, you should be able to:

* '''Identify the major traditional musical instruments and art forms of Ndị Igbo.'''
* '''Contrast the components of traditional Ndị Igbo cosmology.'''
* '''Detail the socio-economic functions of historical practices like traditional marriage, architecture, and the apprenticeship system.'''


== Module 1: Creative Arts and Expressive Traditions ==

=== Music ===
[[File:Udu.jpg|thumb|right|95px|Udu, an Igbo instrument]]
The Igbo people have a melodic and symphonic musical style. Instruments include Ọ̀pì  otherwise known as '''Oja''' a wind instrument similar to the flute, '''igba''', and '''ichaka'''.

Another popular musical form among Igbo people is highlife, which is a fusion of jazz and traditional music and widely popular in West Africa. The modern Igbo highlife is seen in the works of Prince Nico Mbarga, Dr Sir Warrior, Oliver De Coque, Bright Chimezie, Celestine Ukwu,Chief Osita Osadebe, And many others who are some of the greatest Igbo highlife musicians of the twentieth century. There are also other notable Igbo highlife artists, like the Mike Ejeagha, Paulson Kalu, Ali Chukwuma, Ozoemena Nwa Nsugbe.

=== Art ===
Igbo art is known for various types of masquerades, masks, outfits (symbolizing people), animals and abstract conceptions. Igbo art is also known for its bronze castings found in the town of Igbo Ukwu from the 9th century.

&lt;gallery widths=&quot;200&quot; heights=&quot;200&quot; mode=&quot;packed&quot;&gt;
File:Nigeria, igbo, maschera-elmo della società mmuo, xx secolo.jpg|Helmet-mask; 20th century; Indianapolis Museum of Art (USA)
File:Nigeria, igbo, figura femminile per un tempietto, xx secolo.jpg|Female figure for a small temple, 20th century; Indianapolis Museum of Art
File:Igbo brass anklet.jpg|Anklet beaten from a solid brass bar of the type worn by Igbo women. Now in the collection of Wolverhampton Art Gallery. The leg-tube extends approximately 7&amp;nbsp;cm each side of the 35&amp;nbsp;cm disc.
File:Bronze ceremonial vessel in form of a snail shell, 9th century, Igbo-Ukwu, Nigeria.JPG|Bronze ceremonial vessel in form of a snail shell; 9th century; from Igbo-Ukwu; Nigerian National Museum (Lagos, Nigeria)
File:Eze Onyiudo (2).jpg|Eze Onyiudo Masquerade Awka-Etiti
&lt;/gallery&gt;

=== Igbo masks and masquerades ===
There are two basic types of masquerades, visible and invisible. The visible masquerades are meant for the public. They often are more entertaining. Masks used offer a visual appeal for their shapes and forms. In these visible masquerades, performances of harassment, music, dance, and parodies are acted out (Oyeneke 25).

The invisible masquerades take place at night.  Sound is the main tool for them. The masquerader uses his voice to scream so it may be heard throughout the village. The masks used are usually fierce looking and their interpretation is only fully understood by the society's members.  These invisible masquerades call upon a silent village to strike fear in the hearts of those not initiated into their society.


== Module 2: Spiritual Beliefs and Cosmological Frameworks ==

=== Mythology ===
While today many Igbo people are Christian, the traditional ancient Igbo religion is known as Odinani. In the Igbo mythology, which is part of their ancient religion, the supreme God is called Chineke (&quot;the God of creation&quot;); Chineke created the world and everything in it and is associated with all things on Earth. To the ancient Igbo, the cosmos is divided into four complex parts:
 
* OKIKE (Creation)
* ALUSI (Supernatural Forces or Deities)
* MMUO (Spirit)
* UWA (World)

==== Alusi ====
[[File:Complex sculpture Nigeria BM Af1954 23 522 img02.jpg|thumb|alt=A photo of a complex wooden carving of animals, people and spirits laid on each other to about 2 meters in height|Complex wooden carving depicting images of power and daily life, such as horsemen, imported goods, military insignia, Europeans, rifles, wild beasts and masqueraders.]]'''Alusi''', also known as '''Arusi''' or '''Arushi''', are minor deities that are worshiped and served in Igbo mythology. There are a list of many different Alusi that exists within each community and each has its own purpose. When there is no longer need for the deity, it is returned to its source, through the help of a Chief Priest or Dibia, who is aware of the procedure and ensures that its done properly.

==== Mmuo ====
Mmuo simply means spirit. It is either a good and godly spirit (mmuo oma) or it is an evil spirit (mmuo ojo). For example, the Ogbanje spirit is seen as an evil spirit (mmuo ojo) and anyone possessed by this spirit is given spiritual attention. (Spiritual attention means a way of casting out the evil spirit through deliverance (Christian way) or through African Traditional Religion&amp;nbsp; (i.e. digging out his/her '''“iyi uwa”'''. the ATR way)). Ogbanje is an Igbo (Nigeria) term that means a repeater or someone who comes and departs.  Ogbanje is not a bad spirit in Igbo Cosmology. It is a word widely used to describe a kid or teenager who is claimed to die and be born repeatedly by the same person.

=== Osu caste system ===
Osu are a group of people whose ancestors were dedicated to serving in shrines and temples for the deities of the Igbo, and therefore were deemed property of the gods. Relationships and sometimes interactions with Osu were (and to this day, still are) in many cases, forbidden.

To this day being called an ''Osu'' remains a stigma that prevents people's progress and lifestyles.


== Module 3: Social Milestones and Economic Structures ==

=== Yam ===
The yam is very important to the Igbo as it is their staple crop. There are celebrations such as the New yam festival  which are held Every August of Every year for the harvesting of the yam.

The New Yam festival  is celebrated annually to secure a good harvest of the staple crop.  The festival is practiced primarily in Nigeria and other countries in West Africa.

=== Traditional marriage ===
Marriages in Igbo community follow a multi-step process before the bride and groom are proclaimed husband and wife in accordance with local law and tradition.
[[File:Igba nkwu ceremony 04.jpg|alt=Igbo Traditional Marriage|thumb|Traditional Igbo Marriage Attire]]
The traditional marriage is known as &quot;Igbankwu Alumdi&quot; in Igbo land, or wine carrying, since it involves the bride serving up a cup of palm wine to her fiancé. Prior to the wedding, the groom must go to the bride's compound with his father before the Igbankwu day to get the bride's father's consent to marry his daughter. If the bride's father is late, in this case, the bride's brother, uncle or male relative fills in for the bride's late father, as applies to the groom. On the second visit, when kola nuts (oji Igbo) are offered, the two fathers must arrange a price for the bride. In most cases, the bride's price is just symbolic, in addition to other requirements like kola nuts, goats, wine, fowl and so on. Normally, it takes more than one evening until the bride price is agreed upon, after which a feast is served to both parents. When the bride price is paid, another evening is set aside for the ceremony.

During the ceremony, the bride's father fills a cup with palm wine and hands it over to the daughter. Accompanied by her brides maids known as umuagbo nwunye, she then searches for the groom among the crowd of wedding guests to offer him the drink. Once the drink is offered, the bride and groom dance to the bride's father. They kneel before him and he will give them his blessings. After that, the couple dances for a while before taking their seats, then refreshment takes place followed by presentation of gifts, at times a speech from the MC, and then closing prayer and departure.

=== Apprenticeship ===
The Igbo have a unique form of apprenticeship in which either a male family member or a community member will spend time (usually in their teens to their adulthood) with another family, when they work for them. After the time spent with the family, the head of the host household, who is usually the older man who brought the apprentice into his household, will establish  the apprentice by either setting up a business for him or giving money or tools by which to make a living.

This practice was exploited by Europeans, who used this practice as a way of trading in enslaved people. Olaudah Equiano, although stolen from his home, was an Igbo person who was forced into service to an African family.  He said that he felt part of the family, unlike later, when he was shipped to North America and enslaved in the Thirteen Colonies.

The Igbo apprenticeship system is called Imu Ahia or Igba Boy in Igboland. It became more prominent among the Igbos after the Nigerian civil war, in a quest to survive the £20 policy which was proposed by Obafemi Awolowo that only £20 be given to every Biafran citizen to survive on regardless of what they had in the bank before the war and the rest of the money were held by the Nigerian government.

Petty trade was one of the only ways to build back destroyed communities as well as farming, but then, farming required time that was not readily available at that moment. Essentially, most people went into trading.

This Imu-Ahia/Igba Boy model was simple, it works in such a way that business owners would take in younger boys which can be relative, sibling or non-relative from same region, house them and have them work as apprentices in business while learning how it works and the secrets of the business. After the allotted time for the training was reached, 5–8 years’ time, a little graduation ceremony would be held for the '''Nwa Boy''' (the person that learnt the trade). He would also be paid a lump sum for their services over the years, and the money will be used to start a business for the '''Nwa Boy'''.

=== Chieftaincy Title ===
[[File:Igbo ichi marks.jpg|thumb|An Igbo man with ''Ichi'' marks, a sign of rank as an Ozo]]
Highly accomplished men and women are admitted into their noble orders for people of title such as Ndi Ozo or Ndi Nze. These people receive insignia to show their stature. Membership is highly exclusive, and to qualify an individual need to be highly regarded and well-spoken of in the community.

=== Kola nut (Ọjị) ===
[[File:Kola nut.jpg|alt=Kola nut|thumb|Kola nut]]
Kola nut  occupies a unique position in the cultural life of Igbo people. Ọjị is the first thing served to any visitor in an Igbo home. Ọjị is served before an important function begins, be it marriage ceremony, settlement of family disputes or entering into any type of agreement. Ọjị is traditionally broken into pieces by hand, and if the Kola nut breaks into 3 pieces a special celebration is arranged.


== Module 4: Material, Material Culture, Architecture, and Systems of Time ==

=== Igbo Architecture ===
Igbo architecture refers to the architectural styles and building traditions of the Igbo people. The architectural style is closely tied to the Igbo society's culture, beliefs, and social structure. While the architectural style has evolved, traditional Igbo architecture shares some common characteristics such as:

'''Compound layout'''- Igbo architectural traditions often revolve around the concept of a compound which is characterized by an enclosed area encompassing multiple family residences, open central courtyards, verandas, and auxiliary structures. These compounds are meticulously planned and sometimes paved with flat stones to foster communal living and facilitate familial engagements. Additionally, certain compounds feature unique elements like Impluvium houses, Gardens, Moats, and water wells demonstrating the diversity within Igbo architectural practices.

'''Ventilation''' - Igbo architecture integrates strategic placement of openings in buildings to promote cross-ventilation, aiding in regulating indoor temperatures. Employing expansive openings facilitates air circulation, ensuring occupant comfort. Depending on the area with high temperatures and humidity, evaporation of sweat becomes challenging; however, airflow aids this process, enhancing comfort. Moreover, construction practices involve thick walls, thatched roofs, and raised foundations to mitigate environmental challenges. The thick walls maintain cooler interiors in hot weather and warmth during rainy seasons. Thatched roofs provide insulation from direct sunlight, offering shade and contributing to thermal comfort.

'''Shrines and Sacred Spaces'''- Igbo architecture often includes designated spaces in compounds or community areas for ancestral shrines/temples and secret society meeting houses. These spaces are considered sacred and are an essential part of Igbo cultural and religious practices. These sacred structures may vary in design, ranging from simple open-air spaces to more elaborate structures with specific architectural features.

'''Decorative Elements -''' Traditional Igbo architecture often incorporates decorative elements, including painted designs on walls such as [[Uli (design)|uli]], carved wooden door frames, and intricate patterns on ceilings. These decorations may have symbolic or religious significance.

=== Traditional attire ===
Igbo traditional attire varies across regions of Southeastern and south south Nigeria with various cultural significance. 

'''&lt;big&gt;Men&lt;/big&gt;'''

For men, common garments include ''uwe mwuda'' or ''afe ntutu'' ( robe) or ''efe elu'', a basic shirt paired with underneath wrappers or skirts complemented by the ''okpu ozo'' (the feathered red cap), or ''Okpu aji'' (woolen cap), ''ofo'', ''mkpara'' (staff) and ''Akupe'' (handfans) for ceremonial or titled occasions while  loin clothes or waist wrappers were usually worn as casual wears or basic activities like hunting or farming. 

&lt;big&gt;'''Women'''&lt;/big&gt;

[[File:Igbo_woman_wearing_Akwete_obiakwa(double_wrapper)_with_uweobi(blouse)_and_Ichafu_headdress.jpg|thumb|Igbo women’s traditional attire showing Obiakwa(matching double wrappers) made of Akwete George, uweobi (blouse with puffed sleeves) and stiff Ichafu (headdress)]]

Traditional Igbo women's attire comprises many regional and age-based (''Ụmụagbọ'') variant, including the Obiakwa pair of matching wrappers, Uweobi (blouse), and Ịchafụ̀ (head-tie), an elaborate and voluminous headdress traditionally worn by mature women. 

[[File:Igbo_woman_wearing_Isiagu_obiakwa_maiden_attire,_aka_olu(coral_beads)_ngala(head_beads)_and_nza(horsetail).jpg|thumb|A short Obiakwa (wrapper-style) ensemble paired with a fitted blouse, complemented by nza (horsewhisk), ngala (head beads), and other beaded accessories. The textile features  Isiagu motif]]

Younger women may wear shorter Obiakwa wrappers paired with a tubular Uweobi blouse. Traditional adornments include ''aka olu'' (coral beads), ''ngala'' (head beads), and ''mgbaji'' (waist beads), often complemented by other ceremonial accessories like the ''akupe'' (hand fans) and ''nza'' (horsetail whisks). Traditional attire and adornment form many parts of Igbo cultural expressions associated with age, status, ceremony, and identity.

'''&lt;u&gt;Obiakwa&lt;/u&gt;''' 

The ''Obiakwa'' is a traditional women's double- wrapper attire unique to Igbo weaving traditions such as Akwete cloth. It consists of a pair of matching wrappers Descriptions of Akwete weaving note that such wrapper sets were engineered during the weaving process to be worn together.  It is therefore sold in matching pairs.  These wrappers are standardly paired with a blouse called uweobi and the Ichafu headdress. Younger women usually wear shorter ''obiakwa'' waist wrapper sets combined with fitted or tubular blouses or wrappers . These clothings are also complemented with ''ngala'', ''mbaji'' and ''aka'' (beaded accessories) as well ''uli'' body arts. In some regions, The Uli body art was also used to decorate both men and women in the form of lines forming patterns and shapes on the body. 

'''&lt;u&gt;Blouse&lt;/u&gt;''' 

To complete the silhouette, double wrappers(obiakwa) are paired with a Blouse (or ''uweobi''), a traditional fitted blouse. Short ''obiakwa'' styles are usually paired with tubular blouses. 


'''&lt;u&gt;Ichafu&lt;/u&gt;'''

[[File:Igbo_woman_wearing_Joojii_and_Ichafu._Igbo_regality.jpg|thumb|An Igbo woman dressed in traditional attire consisting of a white puff-sleeve Uweobi blouse, a red and gold double George wrapper, and a stiff, elaborately structured Ichafu headdress, accessorized with pearl jewelry.]]

''Ichafu'' is an elaborate head-tie or headdress worn by Igbo women, especially for church services, ceremonies and other social occasions.  It forms part of a broader clothing ensemble that may include wrappers, blouses and jewellery, and is typically tied in volumnious elevated layered styles with large folds and pleats rising above the head. Ichafu is tied with various textiles including synthetic damask, brocade, Akwete and George fabric, which gives it the stiff and highly elaborated look. In Ogadinma, published by Granta, women were described wearing colourful blouses with “expensive ichafu” tied “in layers and pleats until the scarves were piled atop their heads like large plants”. Other dialectical variations for Ichafu is ''Akwaisi'', ''ulari'', ''unari'', ''nsu n'isi'', ''ufu isi'', ''asusu isi'', ''nchafu isi,'' ''Akishi''.   

'''&lt;u&gt;Textiles&lt;/u&gt;'''

Textiles commonly used across Igbo land include ''Isiagu'' (often patterned with the tiger or Lion head motifs), ''Akwete'' and ''Akwaocha'' handwoven clothes, and richly patterned George wrappers.

[[File:Little world, Aichi prefecture - African plaza - Hat of a vassal - Ìgbo people in Nigeria - Collected in 2006.jpg|110px|thumb|left|A traditional Igbo hat made entirely from [[wool]].]]Women carried their babies on their backs with a strip of clothing binding the two with a knot at her chest. This baby carrying technique was and still is practiced by many people groups across Africa, including the Igbo. This method has been modernized in the form of the child carrier.  Both men and women wore wrappers.[[File:Igba nkwu ceremony 07.jpg|thumb|Igba nkwu, Igbo traditional marriage]]

[[File:Igbo Traditional marriage.jpg|thumb|Igbo Traditional Marriage attire]]

=== Calendar (Iguafo Igbo) ===
In the traditional Igbo calendar, a week has 4 days  (''Eke'', ''Orie'', ''Afọ'', ''Nkwọ''), seven weeks make one month, a month has 28 days and there are 13 months in a year. In the last month, an extra day is added. The names of the days have their roots in the mythology of the Kingdom of Nri. It was believed that Eri, the sky-born founder of the Nri kingdom, had gone on a journey to discover the mystery of time. On his journey he had saluted and counted the four days by the names of the spirits that governed them, and so the names of the spirits (''eke'', ''orie'', ''afọ'' and ''Nkwo'') became the days of the week.

{{col-begin}}{{col-2}}
{| class=&quot;wikitable&quot;
!No. || Months (Ọnwa) || Gregorian equivalent
|-
|1 || '''Ọnwa Mbụ''' || (3rd week of February)
|-
|2 || '''Ọnwa Abụa''' || (March)
|-
|3 || '''Ọnwa Ife Eke''' || (April)
|-
|4 || '''Ọnwa Anọ''' || (May)
|-
|5 || '''Ọnwa Agwụ''' || (June)
|-
|6 || '''Ọnwa Ifejiọkụ''' || (July)
|-
|7 || '''Ọnwa Alọm Chi''' || (August to early September)
|-
|8 || '''Ọnwa Ilo Mmụọ''' || (Late September)
|-
|9 || '''Ọnwa Ana''' || (October)
|-
|10 || '''Ọnwa Okike''' || (Early November)
|-
|11 || '''Ọnwa Ajana''' || (Late November)
|-
|12 || '''Ọnwa Ede Ajana''' || (Late November to December)
|-
|13 || '''Ọnwa Ụzọ Alụsị''' || (January to early February)&lt;ref&gt;{{cite book |last1=Onwuejeogwu |first1=M. Angulu |title=An Igbo Civilization: Nri Kingdom &amp; Hegemony |date=1981 |publisher=Ethnographica |isbn=978-978-123-105-6 }}{{page needed|date=January 2024}}&lt;/ref&gt;&lt;ref&gt;{{cite web |url=http://www.free-press-release.com/news/200802/1204305180.html |title=Eze Nri - Igu-Aro Festival - 1008th AD |publisher=Free-Press-Release Inc. |date=February 29, 2008 |access-date=2010-04-06 |archive-date=2009-07-24 |archive-url=https://web.archive.org/web/20090724025818/http://www.free-press-release.com/news/200802/1204305180.html |url-status=dead }}&lt;/ref&gt;
|}

{{col-break}}
An example of a month: '''''Ọnwa Mbụ'''''

{| class=&quot;wikitable&quot;
|-
!Eke || Orie || Afọ || Nkwọ
|-
||||| 1 || 2
|-
|3 || 4 || 5 || 6
|-
|7 || 8 || 9 || 10
|-
|11 || 12 || 13 || 14
|-
|15 || 16 || 17 || 18
|-
|19 || 20 || 21 || 22
|-
|23 || 24 || 25 || 26
|-
|27 || 28||||
|}
{{col-end}}

==== Naming after market days ====
Newborn babies were sometimes named after the day of the week when born. This is no longer the fashion.  Names such as '''Mgbeke''' (maiden [born] on the day of Eke), Mgborie (maiden [born] on the Orie day) are commonly seen among the Igbo people. For males, '''Mgbe''' is replaced by '''Nwa''' or &lt;nowiki&gt;'''&lt;/nowiki&gt;Okoro&lt;nowiki&gt;'''&lt;/nowiki&gt;(Igbo: Child [of]). Examples of this are Solomon Okoronkwo and Nwankwo Kanu, two popular footballers.


[[Category:Igbo culture]]
[[Category:Cultural Anthropology]]
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      <text bytes="21602" sha1="e4i73x8yh4rh02b0s7zkr78s7yrwbjl" xml:space="preserve">== Introduction ==
'''Igbo culture''' are the customs, practices and traditions of the Igbo people of southeastern Nigeria. It consists of ancient practices known as ''Odinala'' ''ndi'' ''igbo'' as well as new concepts added into the Igbo culture either by cultural evolution or by outside influence. These customs and traditions includes the Igbo people's visual art, music and dance forms, as well as their attire, Food, cuisine and language dialects. Because of their various subgroups, the variety of their culture is heightened further.

== Learning Objectives ==
By reviewing this material, you should be able to:

* '''Identify the major traditional musical instruments and art forms of Ndị Igbo.'''
* '''Contrast the components of traditional Ndị Igbo cosmology.'''
* '''Detail the socio-economic functions of historical practices like traditional marriage, architecture, and the apprenticeship system.'''


== Module 1: Creative Arts and Expressive Traditions ==

=== Music ===
[[File:Udu.jpg|thumb|right|95px|Udu, an Igbo instrument]]
The Igbo people have a melodic and symphonic musical style. Instruments include Ọ̀pì  otherwise known as '''Oja''' a wind instrument similar to the flute, '''igba''', and '''ichaka'''.

Another popular musical form among Igbo people is highlife, which is a fusion of jazz and traditional music and widely popular in West Africa. The modern Igbo highlife is seen in the works of Prince Nico Mbarga, Dr Sir Warrior, Oliver De Coque, Bright Chimezie, Celestine Ukwu,Chief Osita Osadebe, And many others who are some of the greatest Igbo highlife musicians of the twentieth century. There are also other notable Igbo highlife artists, like the Mike Ejeagha, Paulson Kalu, Ali Chukwuma, Ozoemena Nwa Nsugbe.

=== Art ===
Igbo art is known for various types of masquerades, masks, outfits (symbolizing people), animals and abstract conceptions. Igbo art is also known for its bronze castings found in the town of Igbo Ukwu from the 9th century.

&lt;gallery widths=&quot;200&quot; heights=&quot;200&quot; mode=&quot;packed&quot;&gt;
File:Nigeria, igbo, maschera-elmo della società mmuo, xx secolo.jpg|Helmet-mask; 20th century; Indianapolis Museum of Art (USA)
File:Nigeria, igbo, figura femminile per un tempietto, xx secolo.jpg|Female figure for a small temple, 20th century; Indianapolis Museum of Art
File:Igbo brass anklet.jpg|Anklet beaten from a solid brass bar of the type worn by Igbo women. Now in the collection of Wolverhampton Art Gallery. The leg-tube extends approximately 7&amp;nbsp;cm each side of the 35&amp;nbsp;cm disc.
File:Bronze ceremonial vessel in form of a snail shell, 9th century, Igbo-Ukwu, Nigeria.JPG|Bronze ceremonial vessel in form of a snail shell; 9th century; from Igbo-Ukwu; Nigerian National Museum (Lagos, Nigeria)
File:Eze Onyiudo (2).jpg|Eze Onyiudo Masquerade Awka-Etiti
&lt;/gallery&gt;

=== Igbo masks and masquerades ===
There are two basic types of masquerades, visible and invisible. The visible masquerades are meant for the public. They often are more entertaining. Masks used offer a visual appeal for their shapes and forms. In these visible masquerades, performances of harassment, music, dance, and parodies are acted out (Oyeneke 25).

The invisible masquerades take place at night.  Sound is the main tool for them. The masquerader uses his voice to scream so it may be heard throughout the village. The masks used are usually fierce looking and their interpretation is only fully understood by the society's members.  These invisible masquerades call upon a silent village to strike fear in the hearts of those not initiated into their society.


== Module 2: Spiritual Beliefs and Cosmological Frameworks ==

=== Mythology ===
While today many Igbo people are Christian, the traditional ancient Igbo religion is known as Odinani. In the Igbo mythology, which is part of their ancient religion, the supreme God is called Chineke (&quot;the God of creation&quot;); Chineke created the world and everything in it and is associated with all things on Earth. To the ancient Igbo, the cosmos is divided into four complex parts:
 
* OKIKE (Creation)
* ALUSI (Supernatural Forces or Deities)
* MMUO (Spirit)
* UWA (World)

==== Alusi ====
[[File:Complex sculpture Nigeria BM Af1954 23 522 img02.jpg|thumb|alt=A photo of a complex wooden carving of animals, people and spirits laid on each other to about 2 meters in height|Complex wooden carving depicting images of power and daily life, such as horsemen, imported goods, military insignia, Europeans, rifles, wild beasts and masqueraders.]]'''Alusi''', also known as '''Arusi''' or '''Arushi''', are minor deities that are worshiped and served in Igbo mythology. There are a list of many different Alusi that exists within each community and each has its own purpose. When there is no longer need for the deity, it is returned to its source, through the help of a Chief Priest or Dibia, who is aware of the procedure and ensures that its done properly.

==== Mmuo ====
Mmuo simply means spirit. It is either a good and godly spirit (mmuo oma) or it is an evil spirit (mmuo ojo). For example, the Ogbanje spirit is seen as an evil spirit (mmuo ojo) and anyone possessed by this spirit is given spiritual attention. (Spiritual attention means a way of casting out the evil spirit through deliverance (Christian way) or through African Traditional Religion&amp;nbsp; (i.e. digging out his/her '''“iyi uwa”'''. the ATR way)). Ogbanje is an Igbo (Nigeria) term that means a repeater or someone who comes and departs.  Ogbanje is not a bad spirit in Igbo Cosmology. It is a word widely used to describe a kid or teenager who is claimed to die and be born repeatedly by the same person.

=== Osu caste system ===
Osu are a group of people whose ancestors were dedicated to serving in shrines and temples for the deities of the Igbo, and therefore were deemed property of the gods. Relationships and sometimes interactions with Osu were (and to this day, still are) in many cases, forbidden.

To this day being called an ''Osu'' remains a stigma that prevents people's progress and lifestyles.


== Module 3: Social Milestones and Economic Structures ==

=== Yam ===
The yam is very important to the Igbo as it is their staple crop. There are celebrations such as the New yam festival  which are held Every August of Every year for the harvesting of the yam.

The New Yam festival  is celebrated annually to secure a good harvest of the staple crop.  The festival is practiced primarily in Nigeria and other countries in West Africa.

=== Traditional marriage ===
Marriages in Igbo community follow a multi-step process before the bride and groom are proclaimed husband and wife in accordance with local law and tradition.
[[File:Igba nkwu ceremony 04.jpg|alt=Igbo Traditional Marriage|thumb|Traditional Igbo Marriage Attire]]
The traditional marriage is known as &quot;Igbankwu Alumdi&quot; in Igbo land, or wine carrying, since it involves the bride serving up a cup of palm wine to her fiancé. Prior to the wedding, the groom must go to the bride's compound with his father before the Igbankwu day to get the bride's father's consent to marry his daughter. If the bride's father is late, in this case, the bride's brother, uncle or male relative fills in for the bride's late father, as applies to the groom. On the second visit, when kola nuts (oji Igbo) are offered, the two fathers must arrange a price for the bride. In most cases, the bride's price is just symbolic, in addition to other requirements like kola nuts, goats, wine, fowl and so on. Normally, it takes more than one evening until the bride price is agreed upon, after which a feast is served to both parents. When the bride price is paid, another evening is set aside for the ceremony.

During the ceremony, the bride's father fills a cup with palm wine and hands it over to the daughter. Accompanied by her brides maids known as umuagbo nwunye, she then searches for the groom among the crowd of wedding guests to offer him the drink. Once the drink is offered, the bride and groom dance to the bride's father. They kneel before him and he will give them his blessings. After that, the couple dances for a while before taking their seats, then refreshment takes place followed by presentation of gifts, at times a speech from the MC, and then closing prayer and departure.

=== Apprenticeship ===
The Igbo have a unique form of apprenticeship in which either a male family member or a community member will spend time (usually in their teens to their adulthood) with another family, when they work for them. After the time spent with the family, the head of the host household, who is usually the older man who brought the apprentice into his household, will establish  the apprentice by either setting up a business for him or giving money or tools by which to make a living.

This practice was exploited by Europeans, who used this practice as a way of trading in enslaved people. Olaudah Equiano, although stolen from his home, was an Igbo person who was forced into service to an African family.  He said that he felt part of the family, unlike later, when he was shipped to North America and enslaved in the Thirteen Colonies.

The Igbo apprenticeship system is called Imu Ahia or Igba Boy in Igboland. It became more prominent among the Igbos after the Nigerian civil war, in a quest to survive the £20 policy which was proposed by Obafemi Awolowo that only £20 be given to every Biafran citizen to survive on regardless of what they had in the bank before the war and the rest of the money were held by the Nigerian government.

Petty trade was one of the only ways to build back destroyed communities as well as farming, but then, farming required time that was not readily available at that moment. Essentially, most people went into trading.

This Imu-Ahia/Igba Boy model was simple, it works in such a way that business owners would take in younger boys which can be relative, sibling or non-relative from same region, house them and have them work as apprentices in business while learning how it works and the secrets of the business. After the allotted time for the training was reached, 5–8 years’ time, a little graduation ceremony would be held for the '''Nwa Boy''' (the person that learnt the trade). He would also be paid a lump sum for their services over the years, and the money will be used to start a business for the '''Nwa Boy'''.

=== Chieftaincy Title ===
[[File:Igbo ichi marks.jpg|thumb|An Igbo man with ''Ichi'' marks, a sign of rank as an Ozo]]
Highly accomplished men and women are admitted into their noble orders for people of title such as Ndi Ozo or Ndi Nze. These people receive insignia to show their stature. Membership is highly exclusive, and to qualify an individual need to be highly regarded and well-spoken of in the community.

=== Kola nut (Ọjị) ===
[[File:Kola nut.jpg|alt=Kola nut|thumb|Kola nut]]
Kola nut  occupies a unique position in the cultural life of Igbo people. Ọjị is the first thing served to any visitor in an Igbo home. Ọjị is served before an important function begins, be it marriage ceremony, settlement of family disputes or entering into any type of agreement. Ọjị is traditionally broken into pieces by hand, and if the Kola nut breaks into 3 pieces a special celebration is arranged.


== Module 4: Material, Material Culture, Architecture, and Systems of Time ==

=== Igbo Architecture ===
Igbo architecture refers to the architectural styles and building traditions of the Igbo people. The architectural style is closely tied to the Igbo society's culture, beliefs, and social structure. While the architectural style has evolved, traditional Igbo architecture shares some common characteristics such as:

'''Compound layout'''- Igbo architectural traditions often revolve around the concept of a compound which is characterized by an enclosed area encompassing multiple family residences, open central courtyards, verandas, and auxiliary structures. These compounds are meticulously planned and sometimes paved with flat stones to foster communal living and facilitate familial engagements. Additionally, certain compounds feature unique elements like Impluvium houses, Gardens, Moats, and water wells demonstrating the diversity within Igbo architectural practices.

'''Ventilation''' - Igbo architecture integrates strategic placement of openings in buildings to promote cross-ventilation, aiding in regulating indoor temperatures. Employing expansive openings facilitates air circulation, ensuring occupant comfort. Depending on the area with high temperatures and humidity, evaporation of sweat becomes challenging; however, airflow aids this process, enhancing comfort. Moreover, construction practices involve thick walls, thatched roofs, and raised foundations to mitigate environmental challenges. The thick walls maintain cooler interiors in hot weather and warmth during rainy seasons. Thatched roofs provide insulation from direct sunlight, offering shade and contributing to thermal comfort.

'''Shrines and Sacred Spaces'''- Igbo architecture often includes designated spaces in compounds or community areas for ancestral shrines/temples and secret society meeting houses. These spaces are considered sacred and are an essential part of Igbo cultural and religious practices. These sacred structures may vary in design, ranging from simple open-air spaces to more elaborate structures with specific architectural features.

'''Decorative Elements -''' Traditional Igbo architecture often incorporates decorative elements, including painted designs on walls such as [[Uli (design)|uli]], carved wooden door frames, and intricate patterns on ceilings. These decorations may have symbolic or religious significance.

=== Traditional attire ===
Igbo traditional attire varies across regions of Southeastern and south south Nigeria with various cultural significance. 

'''&lt;big&gt;Men&lt;/big&gt;'''

For men, common garments include ''uwe mwuda'' or ''afe ntutu'' ( robe) or ''efe elu'', a basic shirt paired with underneath wrappers or skirts complemented by the ''okpu ozo'' (the feathered red cap), or ''Okpu aji'' (woolen cap), ''ofo'', ''mkpara'' (staff) and ''Akupe'' (handfans) for ceremonial or titled occasions while  loin clothes or waist wrappers were usually worn as casual wears or basic activities like hunting or farming. 

&lt;big&gt;'''Women'''&lt;/big&gt;

[[File:Igbo_woman_wearing_Akwete_obiakwa(double_wrapper)_with_uweobi(blouse)_and_Ichafu_headdress.jpg|thumb|Igbo women’s traditional attire showing Obiakwa(matching double wrappers) made of Akwete George, uweobi (blouse with puffed sleeves) and stiff Ichafu (headdress)]]

Traditional Igbo women's attire comprises many regional and age-based (''Ụmụagbọ'') variant, including the Obiakwa pair of matching wrappers, Uweobi (blouse), and Ịchafụ̀ (head-tie), an elaborate and voluminous headdress traditionally worn by mature women. 

[[File:Igbo_woman_wearing_Isiagu_obiakwa_maiden_attire,_aka_olu(coral_beads)_ngala(head_beads)_and_nza(horsetail).jpg|thumb|A short Obiakwa (wrapper-style) ensemble paired with a fitted blouse, complemented by nza (horsewhisk), ngala (head beads), and other beaded accessories. The textile features  Isiagu motif]]

Younger women may wear shorter Obiakwa wrappers paired with a tubular Uweobi blouse. Traditional adornments include ''aka olu'' (coral beads), ''ngala'' (head beads), and ''mgbaji'' (waist beads), often complemented by other ceremonial accessories like the ''akupe'' (hand fans) and ''nza'' (horsetail whisks). Traditional attire and adornment form many parts of Igbo cultural expressions associated with age, status, ceremony, and identity.

'''&lt;u&gt;Obiakwa&lt;/u&gt;''' 

The ''Obiakwa'' is a traditional women's double- wrapper attire unique to Igbo weaving traditions such as Akwete cloth. It consists of a pair of matching wrappers Descriptions of Akwete weaving note that such wrapper sets were engineered during the weaving process to be worn together.  It is therefore sold in matching pairs.  These wrappers are standardly paired with a blouse called uweobi and the Ichafu headdress. Younger women usually wear shorter ''obiakwa'' waist wrapper sets combined with fitted or tubular blouses or wrappers . These clothings are also complemented with ''ngala'', ''mbaji'' and ''aka'' (beaded accessories) as well ''uli'' body arts. In some regions, The Uli body art was also used to decorate both men and women in the form of lines forming patterns and shapes on the body. 

'''&lt;u&gt;Blouse&lt;/u&gt;''' 

To complete the silhouette, double wrappers(obiakwa) are paired with a Blouse (or ''uweobi''), a traditional fitted blouse. Short ''obiakwa'' styles are usually paired with tubular blouses. 


'''&lt;u&gt;Ichafu&lt;/u&gt;'''

[[File:Igbo_woman_wearing_Joojii_and_Ichafu._Igbo_regality.jpg|thumb|An Igbo woman dressed in traditional attire consisting of a white puff-sleeve Uweobi blouse, a red and gold double George wrapper, and a stiff, elaborately structured Ichafu headdress, accessorized with pearl jewelry.]]

''Ichafu'' is an elaborate head-tie or headdress worn by Igbo women, especially for church services, ceremonies and other social occasions.  It forms part of a broader clothing ensemble that may include wrappers, blouses and jewellery, and is typically tied in volumnious elevated layered styles with large folds and pleats rising above the head. Ichafu is tied with various textiles including synthetic damask, brocade, Akwete and George fabric, which gives it the stiff and highly elaborated look. In Ogadinma, published by Granta, women were described wearing colourful blouses with “expensive ichafu” tied “in layers and pleats until the scarves were piled atop their heads like large plants”. Other dialectical variations for Ichafu is ''Akwaisi'', ''ulari'', ''unari'', ''nsu n'isi'', ''ufu isi'', ''asusu isi'', ''nchafu isi,'' ''Akishi''.   

'''&lt;u&gt;Textiles&lt;/u&gt;'''

Textiles commonly used across Igbo land include ''Isiagu'' (often patterned with the tiger or Lion head motifs), ''Akwete'' and ''Akwaocha'' handwoven clothes, and richly patterned George wrappers.

[[File:Little world, Aichi prefecture - African plaza - Hat of a vassal - Ìgbo people in Nigeria - Collected in 2006.jpg|110px|thumb|left|A traditional Igbo hat made entirely from [[wool]].]]Women carried their babies on their backs with a strip of clothing binding the two with a knot at her chest. This baby carrying technique was and still is practiced by many people groups across Africa, including the Igbo. This method has been modernized in the form of the child carrier.  Both men and women wore wrappers.[[File:Igba nkwu ceremony 07.jpg|thumb|Igba nkwu, Igbo traditional marriage]]

[[File:Igbo Traditional marriage.jpg|thumb|Igbo Traditional Marriage attire]]

=== Calendar (Iguafo Igbo) ===
In the traditional Igbo calendar, a week has 4 days  (''Eke'', ''Orie'', ''Afọ'', ''Nkwọ''), seven weeks make one month, a month has 28 days and there are 13 months in a year. In the last month, an extra day is added. The names of the days have their roots in the mythology of the Kingdom of Nri. It was believed that Eri, the sky-born founder of the Nri kingdom, had gone on a journey to discover the mystery of time. On his journey he had saluted and counted the four days by the names of the spirits that governed them, and so the names of the spirits (''eke'', ''orie'', ''afọ'' and ''Nkwo'') became the days of the week.

{{col-begin}}{{col-2}}
{| class=&quot;wikitable&quot;
!No. || Months (Ọnwa) || Gregorian equivalent
|-
|1 || '''Ọnwa Mbụ''' || (3rd week of February)
|-
|2 || '''Ọnwa Abụa''' || (March)
|-
|3 || '''Ọnwa Ife Eke''' || (April)
|-
|4 || '''Ọnwa Anọ''' || (May)
|-
|5 || '''Ọnwa Agwụ''' || (June)
|-
|6 || '''Ọnwa Ifejiọkụ''' || (July)
|-
|7 || '''Ọnwa Alọm Chi''' || (August to early September)
|-
|8 || '''Ọnwa Ilo Mmụọ''' || (Late September)
|-
|9 || '''Ọnwa Ana''' || (October)
|-
|10 || '''Ọnwa Okike''' || (Early November)
|-
|11 || '''Ọnwa Ajana''' || (Late November)
|-
|12 || '''Ọnwa Ede Ajana''' || (Late November to December)
|-
|13 || '''Ọnwa Ụzọ Alụsị''' || (January to early February)&lt;ref&gt;{{cite book |last1=Onwuejeogwu |first1=M. Angulu |title=An Igbo Civilization: Nri Kingdom &amp; Hegemony |date=1981 |publisher=Ethnographica |isbn=978-978-123-105-6 }}{{page needed|date=January 2024}}&lt;/ref&gt;&lt;ref&gt;{{cite web |url=http://www.free-press-release.com/news/200802/1204305180.html |title=Eze Nri - Igu-Aro Festival - 1008th AD |publisher=Free-Press-Release Inc. |date=February 29, 2008 |access-date=2010-04-06 |archive-date=2009-07-24 |archive-url=https://web.archive.org/web/20090724025818/http://www.free-press-release.com/news/200802/1204305180.html |url-status=dead }}&lt;/ref&gt;
|}

{{col-break}}
An example of a month: '''''Ọnwa Mbụ'''''

{| class=&quot;wikitable&quot;
|-
!Eke || Orie || Afọ || Nkwọ
|-
||||| 1 || 2
|-
|3 || 4 || 5 || 6
|-
|7 || 8 || 9 || 10
|-
|11 || 12 || 13 || 14
|-
|15 || 16 || 17 || 18
|-
|19 || 20 || 21 || 22
|-
|23 || 24 || 25 || 26
|-
|27 || 28||||
|}
{{col-end}}

==== Naming after market days ====
Newborn babies were sometimes named after the day of the week when born. This is no longer the fashion.  Names such as '''Mgbeke''' (maiden [born] on the day of Eke), Mgborie (maiden [born] on the Orie day) are commonly seen among the Igbo people. For males, '''Mgbe''' is replaced by '''Nwa''' or &lt;nowiki&gt;'''&lt;/nowiki&gt;Okoro&lt;nowiki&gt;'''&lt;/nowiki&gt;(Igbo: Child [of]). Examples of this are Solomon Okoronkwo and Nwankwo Kanu, two popular footballers.</text>
      <sha1>e4i73x8yh4rh02b0s7zkr78s7yrwbjl</sha1>
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  <page>
    <title>Talk:Umuada</title>
    <ns>1</ns>
    <id>330008</id>
    <revision>
      <id>2813376</id>
      <timestamp>2026-06-07T06:37:04Z</timestamp>
      <contributor>
        <username>Wmbata</username>
        <id>3084293</id>
      </contributor>
      <comment>/* Merging */ new section</comment>
      <origin>2813376</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="252" sha1="4g8pnlbfwpmxypi017vhh8xz8d2hshl" xml:space="preserve">== Merging  ==

Hi @[[User:Atcovi|Atcovi]], i like the idea of merging [[Umuada]] to [[Igbo culture]].
It should be on module 3 [[User:Wmbata|Wmbata]] ([[User talk:Wmbata|discuss]] • [[Special:Contributions/Wmbata|contribs]]) 06:37, 7 June 2026 (UTC)</text>
      <sha1>4g8pnlbfwpmxypi017vhh8xz8d2hshl</sha1>
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