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)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[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; 07:14, 9 June 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)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 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)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===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)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
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)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[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; 07:13, 9 June 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)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 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)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[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; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[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; 08:06, 9 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[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; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[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; 13:59, 11 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)</text>
      <sha1>2no9l6dtlxs1jambf3a8163ytybtwlq</sha1>
    </revision>
    <revision>
      <id>2815421</id>
      <parentid>2815420</parentid>
      <timestamp>2026-06-12T16:30:05Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>/* Draft inactivity policy */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2815421</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38011" sha1="supudwrnp02icmrt27v7d4fgo6gg1nz" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;

== [[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)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[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; 07:14, 9 June 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)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 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)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===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)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
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)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[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; 07:13, 9 June 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)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 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)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 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)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[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; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[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; 08:06, 9 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[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; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[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; 13:59, 11 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)</text>
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  <page>
    <title>Typography</title>
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      <comment>/* Typographers */</comment>
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      <text bytes="2017" sha1="a2rkk6ajduzotcgx2yox0p114p0gydc" xml:space="preserve">[[File:Kerning1.png|thumb|An example of [[w:Kerning]] - formatting the spacings between characters/glyphs]]
'''Typography''' is the art and technique of arranging type to make written language legible, readable, and appealing when displayed. The term also refers to the style, arrangement, and appearance of the letters, numbers, and symbols created by the process. Typography is integral to visual communication and is crucial for graphic design, web design, and desktop publishing. 

== History ==
The history of typography is closely linked to the history of printing.  

Moveable Type (c. 1040), the earliest known system of moveable type was invented in China by Bi Sheng using ceramic materials.  

The Gutenberg Revolution (c. 1440): in Europe, Johannes Gutenberg is credited with inventing the mechanical moveable-type printing press, which began the mass production of books and led to the widespread use of typefaces. His typeface, known as Textura or Blackletter, was based on the elaborate script used by scribes of the time.

== Readings ==
* [[Wikipedia: Typography]]

== Typographers ==
* [[/Jan Tschichold/]]

==Resources on making fonts==
{{w|METAFONT}} is a powerful and complex language that help you develop fonts. It has a sister software in {{w|LaTeX}}, which is used for typestting documents.
*[https://ctan.org/pkg/metafont METAFONT homepage]
*[https://www.latex-project.org/ LaTeX Project]
*[https://tex.stackexchange.com/ TeX at StackExchange]
**[https://tex.stackexchange.com/questions/734849/using-metafont-fonts-in-a-latex-document-guide-and-setup Using METAFONT fonts in a LaTeX document: guide and setup]
*[https://davidcarlisle.github.io/uk-tex-faq/FAQ-useMF.html Getting MetaFont to do what you want]
*[http://metafont.latex.free.fr/ The METAFONT and TeX/LaTeX Resource Page]
*[https://texfaq.org/ TeX FAQ]
*[https://abel.math.harvard.edu/computing/latex/tetex/help/faq/uktug-faq/texfaq_toc.html TeX Frequently Asked Questions]

== References ==
{{reflist}}

[[Category:Typography| ]]</text>
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    <title>Talk:Urantia Book</title>
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      <comment>/* study group */ Reply</comment>
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==study group==
What is &quot;a typical Urantia Book study group&quot;? --[[User:JWSchmidt|JWSchmidt]] 22:28, 2 January 2007 (UTC)
:Urantia Book study groups are a long-standing diverse tradition. See a brief description [http://www.urantia.org/other.html#StudyGroups here]. [[User:CQ|CQ]] 02:26, 3 January 2007 (UTC)
::So does a &quot;diverse tradition&quot; have &quot;typical&quot; format? The [http://www.urantia-iua.org/studygroups_effective.html first page I found] in my google search for such study groups seems to stress the need for keeping out people who are not true believers in the book as a revelation. --[[User:JWSchmidt|JWSchmidt]] 03:28, 3 January 2007 (UTC)
:::It is a gathering of people reading the text. It may be at a home, or church, outdoors, or online. There may be an opening/closing activity. When in-person, there may be socializing and food. Participants usually take turns reading - spontaneously or by assignment - or a facilitator may play an audiobook. They pause to discuss the text as they go along. Some meet regularly and proceed through the entire book, others look at particular selections of the text on a topic. [https://ureadthru.com/ UReadthru] functions like a virtual study group using a chat group and monthly audio/video meetings. For this learning resource to function like a study group, it could perhaps adopt such a chat group format and schedule. Or it could work more asynchronously, like [https://new.ubis.urantia.org/moodle/ Urantia Book International School] and [https://urantiabook.org/learn-with-others/courses-and-seminars Ministers for Christ Michael], using these talk page discussion threads, maybe also with periodic synchronous meetings.
:::[[User:Clehner~enwiki|Clehner~enwiki]] ([[User talk:Clehner~enwiki|discuss]] • [[Special:Contributions/Clehner~enwiki|contribs]]) 03:14, 13 June 2026 (UTC)

:Study groups should encompass everyone. I perused the Urantia Book and felt that it mainly promoted Christianity...so...I edited and condensed it into a 502 page book called [[Urantia United]] - (subtitle) Tapping Into The Mind Of God - For Religious Equality. The UB is absolutely beautifully written...but it was written in 1930s when phrases like &quot;superior races&quot; and inferior races&quot; were acceptable...and Jesus was promoted and favored as the &quot;Son of God&quot; over God's other children...this leads to religious prejudice. 

IMHO, the Urantia Book was written by great intellectuals (normal God-inspired men) in the 1930s when superstitions and prejudice were the norm...other &quot;Holy Books&quot;, written before then, fall into the same category...rewriting them does not diminish them, it enhances their viability. It adds truth and removes superstitions...otherwise, intellectuals and then the normal crowd...will eventually consider them food for the inferior and fables of the older generations...they will be placed in the mythology section. 

A reader's response at TruthBook.com:
&quot;My first sense was that Kurt had plagiarized the Urantia Book, but on a closer examination, I see that Kurt has not altered the words, merely left some concepts out; he has carefully selected a more simplistic message to feed his sheep. For those who have been &quot;damaged&quot; by Christianity, it allows revelation to impact the reader, it speaks to the Thought Adjuster just as adroitly as the original. He has not claimed authorship. He admits editorial license. He acknowledges the original text. 
Remember, Jesus said, &quot;He who is not against us is for us.&quot; And Kurt is definitely not against the revelation. He values it, sees its potential for inaugurating a new dispensation to include all of us, Urantians United&quot;. 
[[User:Kkawohl|Kkawohl]] 00:32, 5 January 2007 (UTC)kkawohl

===Thought Adjuster===
Do you think that [[w:Thought Adjuster]] provides a reasonable description? All this is new to me. --[[User:JWSchmidt|JWSchmidt]] 00:43, 5 January 2007 (UTC)
 
Thought Adjuster is &quot;our spirit&quot;, &quot;our conscience&quot;, &quot;inner voice&quot;, and &quot;divine spark&quot;, among other descriptive phrases.
--[[User:Kkawohl|Kurt]]

==Wingmakers==
I have previously seen the Urantia Book mentioned as an apparent source for the [http://www.wingmakers.com/answersfromjames.html Wingmakers] mythology. Not being very familiar with either, I have no basis upon which to form an opinion. Do you think that ideas were taken from the Urantia Book and used in the Wingmakers myths? Would you be willing to classify Urantia as [[w:New Age|New Age]]? --[[User:JWSchmidt|JWSchmidt]] 02:16, 5 January 2007 (UTC)
----
Parts of Wingmaker's ideas seem to interact with the Urantia Book writings. I don't wish to speak for their classification but the [[Urantia United]] philosophy is along the lines of [http://en.wikipedia.org/wiki/New_Thought New Thought] &quot;The central teaching of New Thought is that thought evolves and unfolds, and thinking creates one's experience of the world. The movement places great emphasis in positive thinking, affirmations, meditation, and prayer...however, they generally have been influenced by a wide range of ideas&quot;. [[User:Kkawohl|Kkawohl]] 04:15, 5 January 2007 (UTC)Kkawohl
----

[[Urantia United]] is designed to support teachers who wish to integrate the principles of religious equality for learning into their theology classroom practice. Assessment advisers, school managers, trainee teachers and researchers may also find these materials helpful. [[User:Kkawohl|Kkawohl]] 23:05, 2 January 2007 (UTC)kkawohl

== [[Topic:Urantia|Department of Urantia Book Studies]] ==

I almost feel bold enough to create a new department, but I'm not familiar enough with the way things work around Wikiversity. Until I get some feedback, I'll just edit ''this page'' in the best fashion I can think of, which might be more or less misguided. [[User:Xaxafrad|Xaxafrad]] 21:11, 14 July 2007 (UTC)
:Hmmm. It would probably be fine from my point of view, but I would call it simply [[Topic:Urantia]] (''see [[Wikiversity:Naming conventions|Naming conventions]]''). I notice the Urantia Book article gets good exposure at [[School:Theology]] under both &quot;Learning resources&quot; and &quot;Wikisource&quot; sections.
 
:Generally, &quot;Topic:&quot; pages are for groups of editors (divisions or departments) managing a number of resources in the main namespace . For example, Wikiversity doesn't even have a [[Topic:Christianity]] even though at least 17 articles in the main namespace could and should be handled by such a group. We have a [[Topic:Scientology]] so... why not? &amp;bull; [[User:CQ|CQ]] 16:39, 7 August 2007 (UTC)
:There is a related organizational discussion at: [[Topic talk:Biblical Studies]]. There is [[Portal:Christianity]]. --[[User:JWSchmidt|JWSchmidt]] 17:34, 7 August 2007 (UTC)

== After the Hiatus... ==

Well, it's been 5 years since I've been around these pages, and it seems to have been a period of hibernation. The overall project (Wikiversity) seems to be going strong, so I'll see if we can breath some new life into the UB corner therein. [[User:Xaxafrad|Xaxafrad]] ([[User talk:Xaxafrad|talk]]) 04:08, 20 December 2012 (UTC)</text>
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      <comment>{{unsigned|Clehner~enwiki|01:41, 9 June 2026}}</comment>
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      <text bytes="4357" sha1="2617nb3yn4ozw3xbwpdnjhr9fsw86bt" xml:space="preserve">What is the real point of this article and why is it here? This is apparently a self-serving presentation by a single individual of his massive abridgement of ''The Urantia Book''. Obviously anyone is entitled to play with their copy of the text of the book in any way they like; however, the presentation of this material in this location (complete with references back to the wikipedia Urantia entry) seems to convey an authority that is altogether absent. Since a naive reader of the UB could become terribly confused and misled by this approach, which is not at all representative of the book itself, I believe the article should be qualified to indicate that it in no way represents the conceptual content of the real book, nor does it reflect any generally accepted approach to abridgement. If this is the product of a seriously minded group of readers, rather than a single overly committed individual, that group should make itself known and open its work to discussion. Absent this, the article should be rewritten to emphasize that this abridged text is NOT ''The Urantia Book'', is NOT representative of the contents of ''The Urantia Book'', and is NOT supported by a community of knowledgeable readers of ''The Urantia Book''. 

This is not to say that a naive reader may not gain spiritual insight from reading this ''Reader's Digest'' version. The selected material is excellent and communicative. But there are very good reasons for not presenting this material in this manner, especially to inexperienced readers. Although it is taken from the Book, this is not very representative of the real Urantia Book. Any why is it in &quot;Wikiversity&quot; anyway? If I receive no replies to this note, I will proceed to edit the entry as I see fit. [[User:68.55.164.69|68.55.164.69]] 21:09, 7 January 2007 (UTC)ldmjr@comcast.net

:Please register a Wikiversity [[Wikiversity:Why create an account|user name]] and edit [[Urantia United|the page]]. By [[wiki|definition]], the content of [[Urantia United|the page]] is open to discussion. This topic is under the wing of the Wikiversity [[School:Theology|School of Theology]] and this applies: &quot;Participants in the Wikiversity School of Theology use rational analysis and argument to discuss, interpret, and teach on any of a myriad a religious topics.&quot; --[[User:JWSchmidt|JWSchmidt]] 21:27, 7 January 2007 (UTC)

::Thank you for the comment. I will follow your advice about registering to edit when ready. I have brought the matter to the attention of The Urantia Book Fellowship and prefer to hear from other interested parties before just digging into what's here. There's no guarantee that what I changed would be an improvement since it's just one person's opinion. It is entirely possible the Fellowship will assign one or more people from their Education Committee to explore this in greater depth. Perhaps they would even take a constructive interest in fostering a group that might emerge around this broad topic. (Should one abridge the Book? Who? What? When? Where? Why? How? etc.) [[User:68.55.164.69|68.55.164.69]] 23:00, 7 January 2007 (UTC)ldmjr@comcast.net

== Urantia Book ==

We also have a new learning resource, '''''[[Urantia Book]]''''' intended to cover the UB in its original unabridged form as a study group and part of [[School:Theology]]. [[User:CQ|CQ]] 00:38, 8 January 2007 (UTC)

== categories ==

Would it be good to add the categories Theology and/or spirituality to this page? [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|talk]]) 17:25, 19 December 2012 (UTC)
:I don't think it would hurt. I'll add them now, if they're not already there. [[User:Xaxafrad|Xaxafrad]] ([[User talk:Xaxafrad|talk]]) 01:56, 17 January 2013 (UTC)

== Name conflict ==

A project was created with the same name, domain name '''urantiaunited.org''' registered in 2018, representing &quot;four major organizations&quot; - Urantia Foundation, The Urantia Book Fellowship, Urantia Association International, and TruthBook.com (Jesusonian Foundation). It makes no mention of [[User:Kkawohl]]'s book or this article. Media linking to it was distributed at the booth at [https://en.wikipedia.org/wiki/Parliament_of_the_World%27s_Religions#2023_Parliament Parliament of the World's Religions]. https://www.urantia.org/news/2023-10/parliament-worlds-religions
{{unsigned|Clehner~enwiki|01:41, 9 June 2026}}</text>
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    <title>Materials Science and Engineering/List of Topics/Origins of Modern Chemistry</title>
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      <comment>Added a missing space in the abstract</comment>
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The history of chemistry is long and convoluted. It begins with the discovery of fire; then metallurgy which allowed purification of metals and the making of alloys, followed by attempts to explain the nature of matter and its transformations through the protoscience of alchemy. Chemistry begins to emerge when the distinction is made between chemistry and alchemy by Robert Boyle in his work The Sceptical Chymist (1661). Chemistry then becomes a full-fledged science when Antoine Lavoisier develops his laws of Conservation of mass, which demands careful measurements and quantitative observations of chemical phenomena. So, while both alchemy and chemistry are concerned with the nature of matter and its transformations, it is only the chemists who apply the scientific method. The history of chemistry is intertwined with the history of thermodynamics, especially through the work of Willard Gibbs.

==Definitions==

In retrospect, the definition of chemistry seems to invariably change per decade, as new discoveries and theories add to the functionality of the science. Shown below are some of the standard definitions used by various noted chemists:

*'''Alchemy''' (330) – the study of the composition of waters, movement, growth, embodying and disembodying, drawing the spirits from bodies and bonding the spirits within bodies (Zosimos of Panopolis).
*'''Chymistry''' (1661) – the subject of the material principles of mixt bodies (Robert Boyle).
*'''Chymistry''' (1663) – a scientific art, by which one learns to dissolve bodies, and draw from them the different substances on their composition, and how to unite them again, and exalt them to an higher perfection (Christopher Glaser).
*'''Chemistry''' (1730) – the art of resolving mixt, compound, or aggregate bodies into their principles; and of composing such bodies from those principles (Georg Ernst Stahl).
*'''Chemistry''' (1837) – the science concerned with the laws and effects of molecular forces (Jean-Baptiste Dumas).
*'''Chemistry''' (1947) – the science of substances: their structure, their properties, and the reactions that change them into other substances (Linus Pauling).
*'''Chemistry''' (1998) – the study of matter and the changes it undergoes (Raymond Chang).
[[Category:Materials Science and Engineering/List of Topics]]</text>
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    <title>Talk:Educational Media Awareness Campaign</title>
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      <comment>/* Where are all images? */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="3813" sha1="qkc6urnftjt1as8t3r4bgmav9vht0z1" xml:space="preserve">==Social sciences==
I realise this is targetted at primary/secondary, but what do you think about a section for [[Portal:Social Sciences|Social Science]] educational pictures? e.g., I'm thinking of images such as [[:Image:Maslow's hierarchy of needs.png]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]]&lt;/small&gt; 07:15, 4 May 2008 (UTC)
: Also, I recommend [[:Image:Structural-Iceberg.svg]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:44, 4 October 2008 (UTC)
::I'm curious:  why? {{unsigned|Jade Knight|16:03, 4 October 2008}}
::: Mainly because it illustrates some commonly-taught concepts in introductory psychology and it's clearly drawn in svg format. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:11, 4 October 2008 (UTC)
::::I see.  You are a Psychology professor, if I recall, so if you think it's a good depiction of basic Psychology, I'll take your word for it.  [[user:Jade Knight|The Jade Knight]] &lt;sup&gt;[[User talk:Jade Knight|(d'viser)]]&lt;/sup&gt; 06:50, 4 October 2008 (UTC)

== Who? ==

Who takes care of all this?  I mean, it's a great idea, but where did it come from?  Who rotates the pictures?  How are they chosen?  [[User:Jade Knight|The Jade Knight]] 06:01, 3 September 2008 (UTC)

: It's all automated; [[User:McCormack]] pretty much created it, I gather. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:23, 3 September 2008 (UTC)

== &quot;20th century&quot; ==
Why do you say &quot;works of art created in the 20th century&quot; -- what's the actual date of creation after which this becomes relevant?  Isn't this also &quot;created since 1923&quot;? [[User:Sj|Sj]] 01:08, 3 November 2009 (UTC)

==Actually being factual about use, especially fair use==
I left this comment on the [[The Great Wave off Kanagawa]] page that was featured as an &quot;educational picture of they day.&quot;  My concern is [[wikipedia:creep | instruction creep]] of the worst sort: fascism.--[[User:John Bessa|John]][[Image:bessa66.png|12px]][[User:John Bessa|Bessa]]&lt;sup&gt;[[User talk:John Bessa|talk]]&lt;/sup&gt; 13:54, 12 April 2010 (UTC)

:The blurb says &quot;''For works of art created in the 20th century, normally you cannot reuse any copies.''&quot;  This statement wholly ignores fair use, as is hence a lie.  

:One can use a copy of a work to discuss the work; I believe that this statement is not so much misconception but part of an orchestrated attempt across the wm to strengthen property, as in intellectual property but also as in real estate, in opposition to the ideas of the public domain that go to social sharing, which is how humanity evolved.

:The wm commons refuses to support fair use, and not for any legal reasons; this fact supports my above statement.

== Chemistry: Absorption/spectroscopy ==

This may be of interest to the rotation boxes.  Does wikuniversity have annotations enabled?
[[File:Spectroscopy_overview.svg|thumb|upright=2]]
[[User:Jkwchui|Jkwchui]] 23:30, 16 January 2011 (UTC)

== Where are all images? ==

Where can one see all the images that appear there?

Also do you have more sources for illustrations and diagrams?&lt;br/&gt;
See [[:Commons:Commons:Village pump#What are free media resources for illustrations?]]. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 20:04, 27 July 2024 (UTC)
: See &quot;Explore this project by subject&quot; at the top of: [[Educational Media Awareness Campaign]]
: More sources are welcome. Do you have a permalink to the discussion you mentioned? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:13, 12 June 2026 (UTC)</text>
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    <title>Binary Stars and Extrasolar Planets</title>
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      <comment>/* Center of Mass */ &quot;...and is closer to the larger masses then the smaller masses.&quot; Changed &quot;then&quot; to &quot;than&quot;.</comment>
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      <text bytes="23126" sha1="8nh1gzwnv0h8e4xgxwd5fno9frhj01y" xml:space="preserve">[[Image:Transiting_planet_HD_209458b.png|thumb|350px|right|An artist conception of the first exosolar planet found to transit the star that it orbits, [[w:HD 209458 b|HD 209458b]].]]
This learning activity utilizes text, imagery, and applet-simulations to introduce the concepts associated with [[w:Binary Star|Binary Star]] systems and the search for [[w:Extrasolar planet|Extrasolar Planets]] (exoplanets for short).  This is a rapidly developing field within [[w:Astronomy|Astronomy]] due to new technology allowing scientists to either directly image or better infer the presence of exosolar planets via gravitational pull, detection of change in visual magnitude, and other methods.  

The activity is separated into three parts to contour the experience into basic, advanced, and mathematical conceptual understanding.  The basic level will introduce the general ideas of what is occurring.  The advanced level will further the conceptual experience to fully understanding the concepts necessary to apply mathematical analysis upon either a binary star system or exoplanet.  The mathematical analysis will introduce [[w:Astrophysics|Astrophysics]] equations in order to give a taste of how scientists analyze the data they collect to aid in the discovery of exoplanets.  Lastly, if you still seek more there is a way that you too can aid in the search for exoplanets without the need for a degree in the field or a large telescope!

When you have completed this activity you should be able to; by level:
Basic: Know terminology and have background-level knowledge of binary systems and exoplanets.
Advanced: Know and understand select techniques pertaining to binary systems and how they can be applied to the search for exoplanets.
Mathematical: Be able to use data to get practical information about either binary stars or exoplanets.


== Basic Concepts: ==


This section looks into the types of binary stars, the [[w:Light curve|light curve]], [[w:Center of Mass|center of mass]], and a simple applet to understand how changing mass and distance causes changes in the orbits of binaries.
[[Image:Sirius_A_and_B_Hubble_photo.jpg|thumb|200px|right|A close-up image of Sirius from [[w:Hubble Space Telescope|Hubble Space Telescope]] revealing it as a Visual Binary with the small [[w:White dwarf|White Dwarf]], Sirius B, to the lower left of Sirius A.]]
'''Types of Binary Stars:'''
#Optical Double: This is actually better used to actually define what constitutes a binary star.  This is not a binary star system and is actually just stars that appear close to each other based upon our vantage point and can be, often are, very far apart.  Thus, the definition of a binary star requires that the stars are gravitationally bound like the Earth and the other planets in our [[w:Solar System|Solar System]] are to the [[w:Sun|Sun]].
#Visual Binary: This describes two gravitationally bound stars that are one of or a combination of the following: bright enough, far enough apart, and/or near enough to be seen separately by high-powered telescopes.  [[w:Albireo|Albireo]], [[w:Mira B|Mira]], and [[w:Sirius|Sirius]] are three examples of visual binaries and have images displaying both stars.  Note that more stars can also be present, [[w:Polaris|Polaris]] (the North Star) is actually a ternary system (three stars) with visual verification. 
#Astrometric Binary: Only one star is visible through current telescopes, but the movement of the star from the gravitational pull of the other star indicates the presence of its unseen companion star. It is a system in which a visible star and a dimmer companion orbit a common centre of mass and detection of such binary by astrmetric means are called astrometric binary.  
#Eclipsing Binary: Eclipsing binary stars means that the star system is oriented, from our vantage point, in such a way that one star passes in front of the other and then later passes behind the other.  This is most notably recognized by a reduction in light due to the one passing in front of the other blocking some or all of the light from the one behind while when both remain visible they both show their entire light output. Algol, β Perseus, also known as the Demon star are the first eclipsing binary.
#Spectroscopic Binary: A [[w:Spectrum|spectrum]] of light at rest produces wavelengths that remain at the same wavelength under all stationary conditions.  However, when moving towards or away from the observer this spectrum shifts.  This method uses the spectra received from stars to note shifts in the position of the bands.  We are then able to know when a star is moving away (shifts the spectrum towards the red end) or towards us (towards blue end).  This is aptly termed [[w:Redshift|red shift]] and [[w:Blue shift|blue shift]].  

All of the above, with the exception of an optical double, can also be applied for exoplanet discovery although the size, mass, and light emission for exoplanets make it considerably more difficult.  Optical doubles are impossible for exoplanets since the overwhelming majority of their light is reflected from the star they orbit.  


=== The Light Curve ===


[[Image:Eclipsing_binary_theorical_light_curve.png|thumb|300px|right|The light, or visual magnitude, curve.]]
As shown, the light curve over the period (the length of the line) of orbit has two drops in [[w:Luminosity|luminosity]].  This would be the data generated by an eclipsing binary star system.  

The first drop is far greater, indicating that it is the passing of the colder, less luminous star in front of the hotter one.  This means that, for every unit area, it is in effect blocking more light.  It does not matter which star, colder or hotter, is larger.

The second drop represents the hotter star passing in front of the cooler one.  It is less because the light being blocked is that of the less luminous star which for every unit area sends less light towards the observer than the hot one. 

The duration of the drops should be approximately the same (not perfectly reflected in this image) as the smaller star disappears at the same rate behind as it blocks the light in front.  Also, the duration reflects the time spent behind or in front of the other star.  The diagonal slopes in and out represent the partial concealment of the star being progressive depending on the duration it takes to fully block the other star.

Again, this can be applied to exoplanets, albeit far more difficult.  Although the lack of light production by a planet assists by decreasing the luminosity to nearly nothing for the gap it makes, the gap of light it actually makes is so much smaller due to tiny radius relative to that of a star that it is nearly unnoticeable to even many modern telescopes.


=== Center of Mass ===


[[Image:Orbit3.gif|thumb|300px|right|Two bodies orbiting a common center of mass.]]
Center of mass is a point at which the combined mass of the two (or more) bodies involved in the rotation act as if they were concentrated at this single point. This point lies between the masses involved, and is closer to the larger masses than the smaller masses. If the system of rotating masses has a transverse velocity the motion can be represented by the motion of the center of mass with this same velocity.  This can be related to the solar system in the sense that the Sun is (essentially, it too rotates some, in actuality, from the pull of the planets) the center of mass by which the planets orbit.  However, when two bodies approach nearer masses this point is drawn out of being located within the heavier body and actually lies at a point in space directly between the two bodies.  It remains equidistance from both stars (in the case of exactly equal mass) as the orbit in their elliptical orbits about it.  The figure shows a center of mass located within the star, but note that they always remain on opposite sides of the center.  More bodies makes the situation far more complex, but ultimately it is the same idea that at any given time the positional motion of all the bodies keeps the central rotation about the center of mass.

=== Application of Basic Ideas ===


We now turn to the Applet to gain an active appreciation for the above concepts.  To keep things simple, the Applet for this section has limited options.  Once you open it, you can see the white dot as the star (the Sun for most of the options) and the blue dot as the planet.  The Applet also greatly exaggerates the movement about the center of mass to exemplify the effect of gravitational binding between the two objects making them both move.  One must be aware of this as the 10 Jupiter setting demonstrates the movement of two near-equal mass bodies whilst in reality ten times the mass of Jupiter is still a very miniscule mass compared to the sun (a mere 0.95%) and would not send the Sun on a crash-course through the solar system as this Applet shows.

Now, let’s run some tests with the Applet[http://www.astro.ubc.ca/~scharein/applets/Sim/xtrasol/XtraSolar.html] (Open in new window if it fails to run in a new tab) and see if you can answer the questions posed correctly.

#Run the simulation for a while with the standard set up of Sun/Jupiter.  Then switch to the Sun/Earth.  What do you notice about the center of mass?  Set to Sun/Jupiter again and observe then switch to two Jupiters, then five Jupiters.  What changes happen with each change in the mass ratio?
#Let us suppose this is a visual binary system and two stars orbiting instead of a star and a planet.  What would we be able to determine about the masses based on watching the rotation?  If one or the other was not visible, would we still know it was a binary system?  What type of binary system would it be?
#Suppose we were looking at this as a spectroscopic binary.  Would we be able to determine anything based on spectra obtained from repeatedly viewing this binary system?
#Let us suppose it’s an eclipsing binary.  If our vantage point was from the bottom of the screen, at what position(s) in the orbit would we see a dip in light?  What about if our vantage point was from the right?

'''Answers:'''
#The center of the mass is inside the Sun when it is using the Earth because the difference of mass is so great, but with Jupiter it was not (remember that in reality it is always within the Sun though, regardless of the planets’ alignment).  As the mass ratio increases the simulation leaves the center of mass at the center of rotation and Jupiter remains at the same location, but the Sun moves further away from the center of mass to maintain the center of mass at a proper distance relative to the mass ratio.
#We would know that the one moving less is more massive.  It would be a spectroscopic binary system because we would only see one star wobbling back and forth in space.
#No, we would not be able to determine anything because it has no motion towards or away from us and just maintains a flattened appearance.  Note, however, that this situation of perfectly perpendicular is very unlikely in practicality and some shift would likely be able to be obtained if the motion was fast enough.
#At the bottom and the top when the two bodies are aligned from our vantage point.  When we were observing from the right, it would be when the stars were on the right and left side of the orbit.  Note that, matter where we placed the vantage point, there would always be two spots in which the light would be reduced and both would be when they came into alignment from our vantage point.

That concludes the basics of binary stars and exoplanets.  We now move on to flesh out more advanced concepts, some of which were alluded to here.


== Advanced Concepts ==

This section looks into the more advanced concepts of: Kepler’s Laws of Planetary Motion, Newton’s version of Kepler’s Third Law, Orientation to Earth, Doppler Shift, and Proportionality.

=== Kepler’s Laws of Planetary Motion ===
[[Image:Kepler laws diagram.svg|thumb|300px|A illustration of Kepler's Laws of Planetary Motion.]]

Kepler’s Laws were created to explain the motion of the planets in the Solar System.  They are based upon Tycho Brahe’s very accurate measuring of the heavens over many years.  They center on the principle of rejecting the geocentric model in favor of the heliocentric model as was necessary to match the data without using epicycles to explain the motion.

They are as follows:
#&quot;The orbit of every planet is an ellipse with the sun at a focus.&quot;
#&quot;A line joining a planet and the sun sweeps out equal areas during equal intervals of time.&quot;
#&quot;The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.&quot;

The first law works on the principle of center of mass.  Kepler determined the orbits were elliptical based upon Tycho’s measurements not fitting the theory using perfect circles which is logical to abide by today when dealing with binaries and exoplanets.

The second law is also useful by relating the speed increase when the objects interacting approach each other and slow when departing.  Velocity is a useful tool towards determining other information about binary stars and stars with exoplanets.

The third law was the first true astrophysical equation.  Although it only applies to objects orbiting the Sun (or other approximately equal mass stars) in its current form it is still useful and becomes greatly more useful when later manipulated by Newton.  Kepler’s third law has become hugely helpful in determining the masses present in binary stars or exoplanets as will be used in the mathematical concepts portion.

The third law in proportional form:

&lt;math&gt; {P^2} \propto  {a^3} &lt;/math&gt;

*P in Years
*a in AU

=== Newton’s Law of Gravitation Applied to Kepler’s Third Law ===


Newton’s Law of Gravitation dictates that all objects in the universe are gravitationally bound to each other.  This is drawn into Kepler’s Laws by the planet exerting a force on the star as well as the star on the planet and also separated the masses so one can apply the property to any objects that are gravitationally bound in a meaningful way (not so distant that the pull has no impact).

Newton’s revised law:

&lt;math&gt;\left({\frac{P}{2\pi}}\right)^2 = {a^3 \over G (M+m)}&lt;/math&gt;

*P in Seconds
*a in Meters
*G is the Gravitational Constant: 6.673x10&lt;sup&gt;-11&lt;/sup&gt;
*M and m in kilograms

This law is commonly used to determine the total mass of visual binaries that then allows extrapolation to large amounts of other data.  

[[Image:Orbital elements.svg|thumb|200px|A view of inclination that would appear flat upon the green plane from Earth.]]

=== Orientation to Earth ===


The orientation to Earth is often known as inclination.  The vast majority of stars provide an orientation of their satellites that is not eclipsing over the center of the star or perfectly upon the celestial sphere.  It is for this reason that when we often are only able to extrapolate a minimum mass when viewing a star’s wobble because we do not know the inclination and, thus, are only able to detect the portion pulling the star on the plane of celestial sphere.  


=== Doppler Shift ===
[[Image:Doppler shift caused by exoplanet.jpeg|thumb|350px|Illustrating the red and blue shift for the observer from an exoplanet.]]

Doppler Shift is the basis for a Spectroscopic Binary system.  It is found by either two separate shifts in spectra or a single shift generated by an unseen companion on the primary star.  It is important because the shifts can be used to find the radial velocity of both stars or the visible one if only one spectrum is observed.  The equation to determine radial velocity is:

&lt;math&gt;{\frac{\Delta \lambda}{\lambda_0}} = {\frac{v_r}{c}}&lt;/math&gt;

c is the speed of light in a vacuum (3x10&lt;sup&gt;8&lt;/sup&gt; m/s)

λ&lt;sub&gt;0&lt;/sub&gt; is the rest wavelength of the spectra

Δλ is the change from the rest wavelength to the measured wavelength

v&lt;sub&gt;r&lt;/sub&gt; is the radial velocity in m/s

If the period is known this can be paired with it to determine the semi-major axis.


=== Proportionality ===


Since all motion involving two objects revolves around a give-and-take relationship there arises the intrinsic relationship between many physical aspects of the two and their behavior with respect to each other.

All of the following relations can be derived from Kepler’s Laws and Doppler Shift and associated mathematical principles.  

&lt;math&gt;{\frac{m_1}{m_2}} = {\frac{r_2}{r_1}} = {\frac{a_2}{a_1}} = {\frac{\alpha_2}{\alpha_1}} = {\frac{v_2}{v_1}} = {\frac{v_2r}{v_1r}} = {\frac{\Delta \lambda_2}{\Delta \lambda_1}}&lt;/math&gt;

Note: Units do not matter as they are ratios and the units cancel.

m is mass

r is the separation distance

a is the length of the semi-major axis

α is the angular separation

v is the velocity

v&lt;sub&gt;r&lt;/sub&gt; is the radial velocity

Δλ is the change in wavelength due to Doppler Shift


=== Application of Advanced Ideas ===


This application section will use a more technical Applet[http://astro.ph.unimelb.edu.au/software/binary/binary.htm] that allows for more intimate manipulation of the model to better experiment with some of the ideas in this section.

# Leaving the model on the default settings, study the layout with radial velocity, the visible light spectrum, the earth view, and the privileged view.  Observe the radial velocity and spectrum.  How do they behave?  What does the negative velocity indicate?  Let the left (red end) rest wavelength be 650 nm.  Calculate the shift in wavelength when the red and blue paths cross and when red peaks in the positive.
# Experiment with the model.  First, adjust the values so that the privileged view is the same as the earth view.  What impact does this have on the Doppler Shift?  Second, adjust it to attain an eclipsing system.  Lastly, make the following changes: a = .8, e = .8, i = 30, w = -45.  Note the radial velocity curve now.  Explain this velocity curve and note the difficulty of being able to understand it if one of the curves did not exist.
# Change the three solar mass star to be .0009535, Jupiter’s mass in terms of solar mass.  Note the Doppler Shift for the star now.  Explain what two changes can be made to the variables (not changing mass) that would aid in discovering this planet.  One will be reflected in the model while the other will not readily do so.
# Formulate a way to prove the concept of center of mass will lie equidistant from both bodies under ideal conditions and test it with the model.
# Using Jupiter’s mass of 1.8986x10&lt;sup&gt;27&lt;/sup&gt; kg, the Sun's mass of 1.9891x10&lt;sup&gt;30&lt;/sup&gt; kg and average distance between them of 7.786x10&lt;sup&gt;11&lt;/sup&gt;  m determine the period of Jupiter in seconds.  Verify this value by using the simplifed version (5.2 AU, ~πx10&lt;sup&gt;7&lt;/sup&gt; s in a year).

'''Answers:'''

# When the velocity is going into the positive the spectrum is revealing a redshift (moving away) and when it is negative it is showing a blueshift (towards).  The negative velocity reflects the movement of the star towards the observer; this is Astronomer's customary view of the motion.  When the paths cross the change is zero because that is the point at which radial velocity from the observer's point of view is zero.  When the red is peaking in the positive (about 27 km/s) the change in wavelength is calculated to be 0.0585 nm. Work shown below.
# Making the inclination 0 makes them match.  This makes the Doppler Shift nothing because they are moving perfectly in the plane of the celestial sphere.  Eclipsing is attained by making the inclination 90 degrees.  The radial velocity curve gets rather difficult to read.  It is important to note that the intersections are still zero so we can be certain of that.  Further, the spikes in speed are when they are the closest as reflected in Kepler's Laws dictating equal area in equal time.  With only one curve it's visibly quite difficult to infer the other and we are unable to be absolutely certain of values generated about such a binary star.
# One change would be to decrease a and thereby bring the period down to a mere 0.03 years which would make observing the star for a couple days reveal an entire period worth of data so that an observer could recognize the fast, but small, wobble of the star.  The other change would be to turn the inclination to 90 degrees so that it would become eclipsing and a dip in light could be spotted from the planet transiting the star.
# Set both masses to be the same.  They will then follow the exact same path around each other.
# Answer worked out below and yes they do match up fairly closely.

'''Doppler Shift worked out:'''

&lt;math&gt;{\frac{\Delta \lambda}{\lambda_0}} = {\frac{v_r}{c}}&lt;/math&gt;

can be rewritten

&lt;math&gt;\Delta \lambda = {\frac{\lambda_0v_r}{c}}&lt;/math&gt;

which equals

&lt;math&gt;\Delta \lambda = {\frac{650nm*27,000m/s}{3x10^8 m/s}}&lt;/math&gt;

and yields

&lt;math&gt;\Delta \lambda = 0.0585&lt;/math&gt;nm.

'''Kepler's Third Law worked out:'''

&lt;math&gt;\left({\frac{P}{2\pi}}\right)^2 = {a^3 \over G (M+m)}&lt;/math&gt;

can be rewritten

&lt;math&gt;P^2 = {4\pi^2a^3 \over G (M+m)}&lt;/math&gt;

which equals

&lt;math&gt;P^2 = {4\pi^2(7.786x10^{11}m)^3 \over 6.673x10^{-11}Nm^2kg^{-2} (1.9891x10^{30}kg+1.8986x10^{27}kg)}&lt;/math&gt;

and yields

&lt;math&gt;P^2 = 1.403x10^{17}&lt;/math&gt;s&lt;sup&gt;2&lt;/sup&gt; → &lt;math&gt;P = 3.745x10^8&lt;/math&gt;s

The simplified version:

&lt;math&gt;P^2 =a^3&lt;/math&gt;

which equals

&lt;math&gt;P^2 =5.2^3&lt;/math&gt;

and yields

&lt;math&gt;P^2 =140.6&lt;/math&gt;yr&lt;sup&gt;2&lt;/sup&gt; → &lt;math&gt;P =11.86&lt;/math&gt;yr

which simplifies to

&lt;math&gt;P = 11.86&lt;/math&gt;yr&lt;math&gt;*\pi x10^7&lt;/math&gt;syr&lt;sup&gt;-1&lt;/sup&gt; &lt;math&gt;= 3.725x10^8&lt;/math&gt;s.

This concludes the advanced concepts that can be associated with binary stars and exoplanets.  To fully explore the nature of these entities continue onto the mathematical concepts that will utilize real stellar data to extrapolate more data about binary stars.

== Mathematical Concepts: ==


To Be Added


== Further Exploration ==

If you are interested in doing further exploration of exoplanets and helping the scientific community in the process you can go to visit Systemic[http://oklo.org/?page_id=33].  The Systemic site has a console to download[http://oklo.org/?page_id=86] which contains radial velocity data for stars which are suspect or known to be supporting exoplanets.  You are able, with the console, to match a curve to the data by adding planets and controlling the planetary mass and distance until it is a good fit for the data accumulated so far.  Upon registering, you are able to submit the curve where the system will compare it to other curves for the same data set generated by other members and, ultimately, as more data becomes available and submissions narrow down possible candidates to the stellar movement your curve fitting may contribute to the verification of the existence of one or more exoplanets orbiting a distant star.  For further explanation of the software, check out their tutorials[http://oklo.org/?page_id=10].
[[Category:Astronomy learning projects]]

== See also ==

* [[Observational astronomy/Extrasolar planet]]</text>
      <sha1>8nh1gzwnv0h8e4xgxwd5fno9frhj01y</sha1>
    </revision>
  </page>
  <page>
    <title>The necessities in Numerical Methods</title>
    <ns>0</ns>
    <id>119778</id>
    <revision>
      <id>2815429</id>
      <parentid>2812789</parentid>
      <timestamp>2026-06-12T18:32:05Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2815429</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="jxjqqoo1yze3zajo4gk57morjwxt8d8" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260608.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

=== Numerical Integration ===
* Trapezoidal Rule
* Simpson's 1/3 Rule
* Romberg Rule
* Gauss-Quadrature Rule
* Adaptive Quadrature
&lt;/br&gt;

=== Roots of a Nonlinear Equation ===
&lt;/br&gt;

=== Optimization ===
&lt;/br&gt;
&lt;/br&gt;

== Matrix Algebra == 

=== Simultaneous Linear Equations === 
* A system of linear equations ([[Media:SystemLinearEq.20240521.pdf |pdf]])


&lt;/br&gt;

=== Gaussian Elimination === 
&lt;/br&gt;

=== LU Decomposition === 
&lt;/br&gt;

=== Cholesky Decomposition === 
&lt;/br&gt;

=== LDL Decomposition === 
&lt;/br&gt;

=== Gauss-Seidel method === 
&lt;/br&gt;

=== Adequacy of Solutions === 
&lt;/br&gt;

=== Eigenvalue and Singular Value ===
&lt;/br&gt;

=== QRD ===
&lt;/br&gt;

=== SVD ===
&lt;/br&gt;

=== Iterative methods ===
&lt;/br&gt;
&lt;/br&gt;

== Regression ==

=== Linear Regression ===
&lt;/br&gt;

=== Non-linear Regression ===
&lt;/br&gt;

=== Linear Least Squares ===
&lt;/br&gt;
&lt;/br&gt; 


== Interpolation ==

=== Polynomial Interpolation ===
&lt;/br&gt;

=== Linear Splines ===
&lt;/br&gt;

=== Piecewise Interpolation ===
&lt;/br&gt;
&lt;/br&gt; 


== Ordinary Differential Equation ==
&lt;/br&gt;

== Partial Differential Equation ==
&lt;/br&gt;

== FEM (Finite Element Method) ==
&lt;/br&gt;
&lt;/br&gt; 


&lt;/br&gt;


== Using Symbolic Package in Octave ==

* Visit http://octave.sourceforge.net/index.html
* Download symbolic-1.0.9.tar.gz

* In Ubuntu, using the Ubuntu Software Center, I installed GiNac and CLN related software and symbolic package for Octave. But it did not properly installed.

* After extracting files from symbolic-1.0.9.tar.gz, I followed the following steps.
 ./configure 
 ./make 
 ./make INSTALL_PATH=/usr/share/octave/packages/3.2/symbolic-1.0.9 

* While doing this, I got an error message related to mkoctfile. So, I used the following command: sudo apt-get install ocatve3.2-headers. Then I was able to install the symbolic packages in the Ubuntu.

== Read some tutorials about symbolic computation ==

* Symbolic Mathematics in Matlab/GNU Octave (http://faraday.elec.uow.edu.au/subjects/annual/ECTE313/Symbolic_Maths.pdf)

* Symbolic Computations (http://www.math.ohiou.edu/courses/math344/lecture7.pdf)

[[Category:Numerical methods]]

== Using SymPy ( a Python library for symbolic mathematics) == 
&lt;/br&gt;


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>jxjqqoo1yze3zajo4gk57morjwxt8d8</sha1>
    </revision>
    <revision>
      <id>2815433</id>
      <parentid>2815429</parentid>
      <timestamp>2026-06-12T18:34:29Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2815433</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="rhzk5gwjaog6kfxw178xqt0em6vlkdl" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260609.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

=== Numerical Integration ===
* Trapezoidal Rule
* Simpson's 1/3 Rule
* Romberg Rule
* Gauss-Quadrature Rule
* Adaptive Quadrature
&lt;/br&gt;

=== Roots of a Nonlinear Equation ===
&lt;/br&gt;

=== Optimization ===
&lt;/br&gt;
&lt;/br&gt;

== Matrix Algebra == 

=== Simultaneous Linear Equations === 
* A system of linear equations ([[Media:SystemLinearEq.20240521.pdf |pdf]])


&lt;/br&gt;

=== Gaussian Elimination === 
&lt;/br&gt;

=== LU Decomposition === 
&lt;/br&gt;

=== Cholesky Decomposition === 
&lt;/br&gt;

=== LDL Decomposition === 
&lt;/br&gt;

=== Gauss-Seidel method === 
&lt;/br&gt;

=== Adequacy of Solutions === 
&lt;/br&gt;

=== Eigenvalue and Singular Value ===
&lt;/br&gt;

=== QRD ===
&lt;/br&gt;

=== SVD ===
&lt;/br&gt;

=== Iterative methods ===
&lt;/br&gt;
&lt;/br&gt;

== Regression ==

=== Linear Regression ===
&lt;/br&gt;

=== Non-linear Regression ===
&lt;/br&gt;

=== Linear Least Squares ===
&lt;/br&gt;
&lt;/br&gt; 


== Interpolation ==

=== Polynomial Interpolation ===
&lt;/br&gt;

=== Linear Splines ===
&lt;/br&gt;

=== Piecewise Interpolation ===
&lt;/br&gt;
&lt;/br&gt; 


== Ordinary Differential Equation ==
&lt;/br&gt;

== Partial Differential Equation ==
&lt;/br&gt;

== FEM (Finite Element Method) ==
&lt;/br&gt;
&lt;/br&gt; 


&lt;/br&gt;


== Using Symbolic Package in Octave ==

* Visit http://octave.sourceforge.net/index.html
* Download symbolic-1.0.9.tar.gz

* In Ubuntu, using the Ubuntu Software Center, I installed GiNac and CLN related software and symbolic package for Octave. But it did not properly installed.

* After extracting files from symbolic-1.0.9.tar.gz, I followed the following steps.
 ./configure 
 ./make 
 ./make INSTALL_PATH=/usr/share/octave/packages/3.2/symbolic-1.0.9 

* While doing this, I got an error message related to mkoctfile. So, I used the following command: sudo apt-get install ocatve3.2-headers. Then I was able to install the symbolic packages in the Ubuntu.

== Read some tutorials about symbolic computation ==

* Symbolic Mathematics in Matlab/GNU Octave (http://faraday.elec.uow.edu.au/subjects/annual/ECTE313/Symbolic_Maths.pdf)

* Symbolic Computations (http://www.math.ohiou.edu/courses/math344/lecture7.pdf)

[[Category:Numerical methods]]

== Using SymPy ( a Python library for symbolic mathematics) == 
&lt;/br&gt;


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>rhzk5gwjaog6kfxw178xqt0em6vlkdl</sha1>
    </revision>
    <revision>
      <id>2815435</id>
      <parentid>2815433</parentid>
      <timestamp>2026-06-12T18:35:16Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2815435</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="gazlpi3du389xeusf4nu3gjt8u2zay5" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260610.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

=== Numerical Integration ===
* Trapezoidal Rule
* Simpson's 1/3 Rule
* Romberg Rule
* Gauss-Quadrature Rule
* Adaptive Quadrature
&lt;/br&gt;

=== Roots of a Nonlinear Equation ===
&lt;/br&gt;

=== Optimization ===
&lt;/br&gt;
&lt;/br&gt;

== Matrix Algebra == 

=== Simultaneous Linear Equations === 
* A system of linear equations ([[Media:SystemLinearEq.20240521.pdf |pdf]])


&lt;/br&gt;

=== Gaussian Elimination === 
&lt;/br&gt;

=== LU Decomposition === 
&lt;/br&gt;

=== Cholesky Decomposition === 
&lt;/br&gt;

=== LDL Decomposition === 
&lt;/br&gt;

=== Gauss-Seidel method === 
&lt;/br&gt;

=== Adequacy of Solutions === 
&lt;/br&gt;

=== Eigenvalue and Singular Value ===
&lt;/br&gt;

=== QRD ===
&lt;/br&gt;

=== SVD ===
&lt;/br&gt;

=== Iterative methods ===
&lt;/br&gt;
&lt;/br&gt;

== Regression ==

=== Linear Regression ===
&lt;/br&gt;

=== Non-linear Regression ===
&lt;/br&gt;

=== Linear Least Squares ===
&lt;/br&gt;
&lt;/br&gt; 


== Interpolation ==

=== Polynomial Interpolation ===
&lt;/br&gt;

=== Linear Splines ===
&lt;/br&gt;

=== Piecewise Interpolation ===
&lt;/br&gt;
&lt;/br&gt; 


== Ordinary Differential Equation ==
&lt;/br&gt;

== Partial Differential Equation ==
&lt;/br&gt;

== FEM (Finite Element Method) ==
&lt;/br&gt;
&lt;/br&gt; 


&lt;/br&gt;


== Using Symbolic Package in Octave ==

* Visit http://octave.sourceforge.net/index.html
* Download symbolic-1.0.9.tar.gz

* In Ubuntu, using the Ubuntu Software Center, I installed GiNac and CLN related software and symbolic package for Octave. But it did not properly installed.

* After extracting files from symbolic-1.0.9.tar.gz, I followed the following steps.
 ./configure 
 ./make 
 ./make INSTALL_PATH=/usr/share/octave/packages/3.2/symbolic-1.0.9 

* While doing this, I got an error message related to mkoctfile. So, I used the following command: sudo apt-get install ocatve3.2-headers. Then I was able to install the symbolic packages in the Ubuntu.

== Read some tutorials about symbolic computation ==

* Symbolic Mathematics in Matlab/GNU Octave (http://faraday.elec.uow.edu.au/subjects/annual/ECTE313/Symbolic_Maths.pdf)

* Symbolic Computations (http://www.math.ohiou.edu/courses/math344/lecture7.pdf)

[[Category:Numerical methods]]

== Using SymPy ( a Python library for symbolic mathematics) == 
&lt;/br&gt;


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>gazlpi3du389xeusf4nu3gjt8u2zay5</sha1>
    </revision>
    <revision>
      <id>2815437</id>
      <parentid>2815435</parentid>
      <timestamp>2026-06-12T18:37:31Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2815437</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="kfi7c69268023wv9le02mcqorueft8s" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260611.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

=== Numerical Integration ===
* Trapezoidal Rule
* Simpson's 1/3 Rule
* Romberg Rule
* Gauss-Quadrature Rule
* Adaptive Quadrature
&lt;/br&gt;

=== Roots of a Nonlinear Equation ===
&lt;/br&gt;

=== Optimization ===
&lt;/br&gt;
&lt;/br&gt;

== Matrix Algebra == 

=== Simultaneous Linear Equations === 
* A system of linear equations ([[Media:SystemLinearEq.20240521.pdf |pdf]])


&lt;/br&gt;

=== Gaussian Elimination === 
&lt;/br&gt;

=== LU Decomposition === 
&lt;/br&gt;

=== Cholesky Decomposition === 
&lt;/br&gt;

=== LDL Decomposition === 
&lt;/br&gt;

=== Gauss-Seidel method === 
&lt;/br&gt;

=== Adequacy of Solutions === 
&lt;/br&gt;

=== Eigenvalue and Singular Value ===
&lt;/br&gt;

=== QRD ===
&lt;/br&gt;

=== SVD ===
&lt;/br&gt;

=== Iterative methods ===
&lt;/br&gt;
&lt;/br&gt;

== Regression ==

=== Linear Regression ===
&lt;/br&gt;

=== Non-linear Regression ===
&lt;/br&gt;

=== Linear Least Squares ===
&lt;/br&gt;
&lt;/br&gt; 


== Interpolation ==

=== Polynomial Interpolation ===
&lt;/br&gt;

=== Linear Splines ===
&lt;/br&gt;

=== Piecewise Interpolation ===
&lt;/br&gt;
&lt;/br&gt; 


== Ordinary Differential Equation ==
&lt;/br&gt;

== Partial Differential Equation ==
&lt;/br&gt;

== FEM (Finite Element Method) ==
&lt;/br&gt;
&lt;/br&gt; 


&lt;/br&gt;


== Using Symbolic Package in Octave ==

* Visit http://octave.sourceforge.net/index.html
* Download symbolic-1.0.9.tar.gz

* In Ubuntu, using the Ubuntu Software Center, I installed GiNac and CLN related software and symbolic package for Octave. But it did not properly installed.

* After extracting files from symbolic-1.0.9.tar.gz, I followed the following steps.
 ./configure 
 ./make 
 ./make INSTALL_PATH=/usr/share/octave/packages/3.2/symbolic-1.0.9 

* While doing this, I got an error message related to mkoctfile. So, I used the following command: sudo apt-get install ocatve3.2-headers. Then I was able to install the symbolic packages in the Ubuntu.

== Read some tutorials about symbolic computation ==

* Symbolic Mathematics in Matlab/GNU Octave (http://faraday.elec.uow.edu.au/subjects/annual/ECTE313/Symbolic_Maths.pdf)

* Symbolic Computations (http://www.math.ohiou.edu/courses/math344/lecture7.pdf)

[[Category:Numerical methods]]

== Using SymPy ( a Python library for symbolic mathematics) == 
&lt;/br&gt;


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>kfi7c69268023wv9le02mcqorueft8s</sha1>
    </revision>
  </page>
  <page>
    <title>VHDL programming in plain view</title>
    <ns>0</ns>
    <id>121359</id>
    <revision>
      <id>2815439</id>
      <parentid>2812680</parentid>
      <timestamp>2026-06-12T18:52:59Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2815439</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="kbwh90zyh9s15m4xbqsm5aevmwd6c96" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== Flip Flop and Latch ==
* FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]])
* Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]])
* Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]])
* Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260608.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]])
* Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]])
* Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]])
* Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== Structure ====
* Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]])
* Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]])
* Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== Function ====
* Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]])
* Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]])
* Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
[[Category:FPGA]]</text>
      <sha1>kbwh90zyh9s15m4xbqsm5aevmwd6c96</sha1>
    </revision>
    <revision>
      <id>2815474</id>
      <parentid>2815439</parentid>
      <timestamp>2026-06-13T11:48:12Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2815474</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="n598hd0qyqcamo0tykxbjuiknrbdz55" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== Flip Flop and Latch ==
* FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]])
* Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]])
* Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]])
* Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260609.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]])
* Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]])
* Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]])
* Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== Structure ====
* Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]])
* Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]])
* Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== Function ====
* Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]])
* Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]])
* Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
[[Category:FPGA]]</text>
      <sha1>n598hd0qyqcamo0tykxbjuiknrbdz55</sha1>
    </revision>
    <revision>
      <id>2815476</id>
      <parentid>2815474</parentid>
      <timestamp>2026-06-13T11:49:08Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2815476</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="se47zwv8kh640vfn3lw2t4q7jfv8vfr" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== Flip Flop and Latch ==
* FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]])
* Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]])
* Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]])
* Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260610.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]])
* Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]])
* Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]])
* Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== Structure ====
* Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]])
* Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]])
* Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== Function ====
* Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]])
* Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]])
* Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
[[Category:FPGA]]</text>
      <sha1>se47zwv8kh640vfn3lw2t4q7jfv8vfr</sha1>
    </revision>
    <revision>
      <id>2815478</id>
      <parentid>2815476</parentid>
      <timestamp>2026-06-13T11:50:00Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2815478</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="7dy9g3ks2lxbfavgjvc8th3r8bdlyof" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== Flip Flop and Latch ==
* FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]])
* Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]])
* Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]])
* Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260611.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]])
* Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]])
* Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]])
* Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== Structure ====
* Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]])
* Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]])
* Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== Function ====
* Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]])
* Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]])
* Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
[[Category:FPGA]]</text>
      <sha1>7dy9g3ks2lxbfavgjvc8th3r8bdlyof</sha1>
    </revision>
    <revision>
      <id>2815480</id>
      <parentid>2815478</parentid>
      <timestamp>2026-06-13T11:50:46Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2815480</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="bfsnohelbue8j5d9vgtbk84w2i7jfvq" xml:space="preserve">&lt;!----------------------------------------------------------------------&gt;

== Flip Flop and Latch ==
* FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]])
* Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]])
* Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]])
* Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]])
&lt;br&gt; 

== Versions of VHDL ==
* VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]])
* VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]])
&lt;br&gt;

== Basic Features of VHDL ==
==== Data ====
* Data Objects ([[Media:Data.Object.1A.20260612.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]])
* Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]])
* Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]])
* Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]])
&lt;br&gt;

==== Signals &amp; Variables ====
* Signals &amp; Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]])
* Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]])
* Concurrent &amp; Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]])
* Inertial &amp; Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]])
* Simulation &amp; Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]])


&lt;br&gt;

==== Structure ====
* Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]])
* Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]])
* Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]])
&lt;/br&gt;

==== Entity and Architecture ====

&lt;br&gt; 
==== Block Statement ====

&lt;br&gt; 
==== Process Statement ====

&lt;br&gt;
==== Operators ====

&lt;br&gt;
==== Assignment Statement ====

&lt;br&gt; 
==== Concurrent Statement ====

&lt;br&gt; 
==== Sequential Control Statement ====

&lt;br&gt;
==== Function ====
* Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]])
* Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]])
* Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]])

&lt;br&gt;
==== Procedure ====


&lt;br&gt;
==== Package ====
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:VHDL]]
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      <text bytes="74037" sha1="dt2zq70ytvzzwjoo2awpcfvz0ntff1n" xml:space="preserve">== Archives ==
* [[Outreach:GLAM/Newsletter/Archives]]

== ''This Month in GLAM'': January 2025 ==

{| style=&quot;width:100%;&quot;
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{| align=&quot;center&quot;
|-
| style=&quot;text-align: center;&quot; | [[File:This Month in GLAM logo 2018.png|350px|center|link=outreach:GLAM/Newsletter]]&lt;br /&gt;
&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/January 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue I, January 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
| &lt;span style=&quot;font-size:12pt; font-family:Times New Roman;&quot;&gt; '''&lt;u&gt;Headlines&lt;/u&gt;'''&lt;/span&gt;
|- style=&quot;font-size:10pt; font-family:Times New Roman; text-align:center;&quot;
| &lt;div style=&quot;text-align:left; column-count:2; column-width:28em; vertical-align:top;&quot;&gt;

* [[outreach:GLAM/Newsletter/January 2025/Contents/Africa report|Africa report]]: Guinea-Bissau Heritage from Commons to the World
* [[outreach:GLAM/Newsletter/January 2025/Contents/Albania report|Albania report]]: Wikimedian in Residence at the Qemal Baholli Public Library in Elbasan (October - December 2024)
* [[outreach:GLAM/Newsletter/January 2025/Contents/Aruba report|Aruba report]]: Wikipedia on Aruba project has officially begun!
* [[outreach:GLAM/Newsletter/January 2025/Contents/Brazil report|Brazil report]]: Wiki Loves Maranhão
* [[outreach:GLAM/Newsletter/January 2025/Contents/Germany report|Germany report]]: Exploring Wikidata &amp; Building Community for Cultural Heritage Professionals
* [[outreach:GLAM/Newsletter/January 2025/Contents/Indonesia report|Indonesia report]]: Celebrating Public Domain Day 2025 in Indonesia
* [[outreach:GLAM/Newsletter/January 2025/Contents/Italy report|Italy report]]: New Wikimedia Italia Grant for GLAMs
* [[outreach:GLAM/Newsletter/January 2025/Contents/Netherlands report|Netherlands report]]: 3 Million Dutch Cultural Heritage Images in Commons &amp; 400,000 RCE images now in higher resolution &amp; Usage of DBNL in Dutch Wikipedia articles
* [[outreach:GLAM/Newsletter/January 2025/Contents/New Zealand report|New Zealand report]]: Student led Edit-a-thon
* [[outreach:GLAM/Newsletter/January 2025/Contents/Poland report|Poland report]]: GLAM-Wiki 2024 in Poland: Achievements, Collaborations, and Impact
* [[outreach:GLAM/Newsletter/January 2025/Contents/Serbia report|Serbia report]]: Wikimedia Serbia: Advancing GLAM collaborations and digital heritage
* [[outreach:GLAM/Newsletter/January 2025/Contents/Switzerland report|Switzerland report]]: Swiss GLAM Programme
* [[outreach:GLAM/Newsletter/January 2025/Contents/UK report|UK report]]: Cairo Geniza
* [[outreach:GLAM/Newsletter/January 2025/Contents/USA report|USA report]]: Wikipedia Day
* [[outreach:GLAM/Newsletter/January 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: BHL-Wiki Working Group January monthly highlights
* [[outreach:GLAM/Newsletter/January 2025/Contents/Special story|Special story]]: Join the Global GLAM Call – Tuesday, February 11th, 13:30 UTC!
* [[outreach:GLAM/Newsletter/January 2025/Contents/Memory of the World report|Memory of the World report]]: To the front page!
* [[outreach:GLAM/Newsletter/January 2025/Contents/Wikidata report|Wikidata report]]: Wikidata at WikiLibCon 2025
* [[outreach:GLAM/Newsletter/January 2025/Contents/Events|Calendar]]: February's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/January 2025/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 23:23, 9 February 2025 (UTC)&lt;/div&gt;
&lt;!-- Message sent by User:Romaine@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/GLAM&amp;oldid=28125057 --&gt;

== ''This Month in GLAM'': February 2025 ==

{| style=&quot;width:100%;&quot;
| valign=&quot;top&quot; align=&quot;center&quot; style=&quot;border:1px gray solid; padding:1em;&quot; |
{| align=&quot;center&quot;
|-
| style=&quot;text-align: center;&quot; | [[File:This Month in GLAM logo 2018.png|350px|center|link=outreach:GLAM/Newsletter]]&lt;br /&gt;
&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/February 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue II, February 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
| &lt;span style=&quot;font-size:12pt; font-family:Times New Roman;&quot;&gt; '''&lt;u&gt;Headlines&lt;/u&gt;'''&lt;/span&gt;
|- style=&quot;font-size:10pt; font-family:Times New Roman; text-align:center;&quot;
| &lt;div style=&quot;text-align:left; column-count:2; column-width:28em; vertical-align:top;&quot;&gt;

* [[outreach:GLAM/Newsletter/February 2025/Contents/Albania report|Albania report]]: Celebrating the English Wikipedia’s Birthday in Albania!
* [[outreach:GLAM/Newsletter/February 2025/Contents/Belgium report|Belgium report]]: Belgium Public Domain Day and Dance Heritage
* [[outreach:GLAM/Newsletter/February 2025/Contents/Brazil report|Brazil report]]: Wiki and COP30 in the Amazon rainforest
* [[outreach:GLAM/Newsletter/February 2025/Contents/Germany report|Germany report]]: GLAM digital and seminar on Jewish life 
* [[outreach:GLAM/Newsletter/February 2025/Contents/Italy report|Italy report]]: GLAM call and Progetto cultura
* [[outreach:GLAM/Newsletter/February 2025/Contents/Netherlands report|Netherlands report]]: [GLAM metrics] Usage of Delpher in Dutch Wikipedia articles
* [[outreach:GLAM/Newsletter/February 2025/Contents/New Zealand report|New Zealand report]]: Wikipedia podcast episode, Trilepidea newsletter article, and the Wikipedian at Large 
* [[outreach:GLAM/Newsletter/February 2025/Contents/Macedonia report|North Macedonia report]]: Wikimedia MKD GLAM program for 2025
* [[outreach:GLAM/Newsletter/February 2025/Contents/Poland report|Poland report]]: What's up in GLAM in Poland
* [[outreach:GLAM/Newsletter/February 2025/Contents/Switzerland report|Switzerland report]]: Swiss GLAM Programme
* [[outreach:GLAM/Newsletter/February 2025/Contents/UK report|UK report]]: Islamic and Jewish history
* [[outreach:GLAM/Newsletter/February 2025/Contents/Ukraine report|Ukraine report]]: GLAM news from Ukraine – events for libraries, #1Lib1Ref, launch of a larger GLAM product 
* [[outreach:GLAM/Newsletter/February 2025/Contents/USA report|USA report]]: February meetings
* [[outreach:GLAM/Newsletter/February 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: BHL-Wiki Working Group February monthly highlights
* [[outreach:GLAM/Newsletter/February 2025/Contents/AvoinGLAM report|AvoinGLAM report]]: Connecting Media Art Archives
* [[outreach:GLAM/Newsletter/February 2025/Contents/Memory of the World report|Memory of the World report]]: 5.1 million image views
* [[outreach:GLAM/Newsletter/February 2025/Contents/Wikidata report|Wikidata report]]: Wikidata event: Data Reuse Days 2025
* [[outreach:GLAM/Newsletter/February 2025/Contents/Wikisource report|Wikisource report]]: Wikisource Conference 2025
* [[outreach:GLAM/Newsletter/February 2025/Contents/Wikimedia and Libraries User Group report|Wikimedia and Libraries User Group report]]: Wikimedia + Libraries International Convention 2025
* [[outreach:GLAM/Newsletter/February 2025/Contents/Events|Calendar]]: March's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/February 2025/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 16:51, 10 March 2025 (UTC)&lt;/div&gt;
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== ''This Month in GLAM'': March 2025 ==

{| style=&quot;width:100%;&quot;
| valign=&quot;top&quot; align=&quot;center&quot; style=&quot;border:1px gray solid; padding:1em;&quot; |
{| align=&quot;center&quot;
|-
| style=&quot;text-align: center;&quot; | [[File:This Month in GLAM logo 2018.png|350px|center|link=outreach:GLAM/Newsletter]]&lt;br /&gt;
&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/March 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue III, March 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
| &lt;span style=&quot;font-size:12pt; font-family:Times New Roman;&quot;&gt; '''&lt;u&gt;Headlines&lt;/u&gt;'''&lt;/span&gt;
|- style=&quot;font-size:10pt; font-family:Times New Roman; text-align:center;&quot;
| &lt;div style=&quot;text-align:left; column-count:2; column-width:28em; vertical-align:top;&quot;&gt;

* [[outreach:GLAM/Newsletter/March 2025/Contents/Albania report|Albania report]]: WikiGap in Albania 2025, and essential initiatives for free knowledge
* [[outreach:GLAM/Newsletter/March 2025/Contents/Argentina report|Argentina report]]: Archives and Human Rights activities
* [[outreach:GLAM/Newsletter/March 2025/Contents/Belgium report|Belgium report]]: International Women's Day, Wikipedia officially recognised as Digital Public Good and invitation for the General Assembly
* [[outreach:GLAM/Newsletter/March 2025/Contents/Brazil report|Brazil report]]: Every Book Its Reader is coming
* [[outreach:GLAM/Newsletter/March 2025/Contents/Czech Republic report|Czech Republic report]]: New Horizons of Czech GLAM Partnerships in 2025
* [[outreach:GLAM/Newsletter/March 2025/Contents/France report|France report]]: Archivist Forum 2025
* [[outreach:GLAM/Newsletter/March 2025/Contents/Italy report|Italy report]]: Ten winners of Wikimedia Italia 2025 GLAM call
* [[outreach:GLAM/Newsletter/March 2025/Contents/Netherlands report|Netherlands report]]: International Womensday; New Wikimedian in Residence at Maastricht University
* [[outreach:GLAM/Newsletter/March 2025/Contents/New Zealand report|New Zealand report]]: What's on at Auckland Museum, and International Women's Day at University of Otago 
* [[outreach:GLAM/Newsletter/March 2025/Contents/Nigeria report|Nigeria report]]: GLAM in Africa, a Nigerian narrative in knowledge decolonization a case study of Benin City
* [[outreach:GLAM/Newsletter/March 2025/Contents/Poland report|Poland report]]: What's up in GLAM in Poland
* [[outreach:GLAM/Newsletter/March 2025/Contents/Switzerland report|Switzerland report]]: Switzerland report
* [[outreach:GLAM/Newsletter/March 2025/Contents/UK report|UK report]]: Gold in Bengali and Diversity in Arabic
* [[outreach:GLAM/Newsletter/March 2025/Contents/USA report|USA report]]: Women's History month
* [[outreach:GLAM/Newsletter/March 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: BHL-Wiki Working Group March monthly highlights
* [[outreach:GLAM/Newsletter/March 2025/Contents/Memory of the World report|Memory of the World report]]: Manuscripts of Mali
* [[outreach:GLAM/Newsletter/March 2025/Contents/Structured Data on Wikimedia Commons report|Structured Data on Wikimedia Commons report]]: Creating an OpenRefine Wikimedia Group
* [[outreach:GLAM/Newsletter/March 2025/Contents/Events|Calendar]]: April's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/March 2025/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 09:00, 9 April 2025 (UTC)&lt;/div&gt;
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== ''This Month in GLAM'': April 2025 ==

{| style=&quot;width:100%;&quot;
| valign=&quot;top&quot; align=&quot;center&quot; style=&quot;border:1px gray solid; padding:1em;&quot; |
{| align=&quot;center&quot;
|-
| style=&quot;text-align: center;&quot; | [[File:This Month in GLAM logo 2018.png|350px|center|link=outreach:GLAM/Newsletter]]&lt;br /&gt;
&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/April 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue IV, April 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
| &lt;span style=&quot;font-size:12pt; font-family:Times New Roman;&quot;&gt; '''&lt;u&gt;Headlines&lt;/u&gt;'''&lt;/span&gt;
|- style=&quot;font-size:10pt; font-family:Times New Roman; text-align:center;&quot;
| &lt;div style=&quot;text-align:left; column-count:2; column-width:28em; vertical-align:top;&quot;&gt;

* [[outreach:GLAM/Newsletter/April 2025/Contents/Albania report|Albania report]]: Wikipedia Event for International Roma Day 2025
* [[outreach:GLAM/Newsletter/April 2025/Contents/Australia report|Australia report]]: Highlighting Feminist and women's histories and a Wiki Day at the South Australian Museum
* [[outreach:GLAM/Newsletter/April 2025/Contents/Catalan areas report|Catalan areas report]]: Campaign to document the 2025 Falla monuments in Valencia
* [[outreach:GLAM/Newsletter/April 2025/Contents/Italy report|Italy report]]: Wikidata and Research
* [[outreach:GLAM/Newsletter/April 2025/Contents/Netherlands report|Netherlands report]]: Open Collection Highlights
* [[outreach:GLAM/Newsletter/April 2025/Contents/New Zealand report|New Zealand report]]: Women in Architecture, BHL, and the Commons Workflow
* [[outreach:GLAM/Newsletter/April 2025/Contents/Nigeria report|Nigeria report]]: Strengthening Cultural Heritage through Partnerships and Knowledge Sharing: Insights from World Heritage Day in Nigeria
* [[outreach:GLAM/Newsletter/April 2025/Contents/Macedonia report|North Macedonia report]]: Wikimedia MKD's new GLAM collaborations and activities 
* [[outreach:GLAM/Newsletter/April 2025/Contents/Poland report|Poland report]]: What's up in GLAM in Poland
* [[outreach:GLAM/Newsletter/April 2025/Contents/Portugal report|Portugal report]]: Scholarships and Call for Sessions Proposals for GLAM Wiki 2025 will open soon: Stay tuned!
* [[outreach:GLAM/Newsletter/April 2025/Contents/Serbia report|Serbia report]]: GLAM Highlights from Serbia
* [[outreach:GLAM/Newsletter/April 2025/Contents/Switzerland report|Switzerland report]]: FemNetzCon 2025; GLAM Meeting Biel/Bienne; Digi Archive
* [[outreach:GLAM/Newsletter/April 2025/Contents/UK report|UK report]]: Mapping Museums / Art in Arabic
* [[outreach:GLAM/Newsletter/April 2025/Contents/USA report|USA report]]: April meetings
* [[outreach:GLAM/Newsletter/April 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: BHL-Wiki Working Group April monthly highlights
* [[outreach:GLAM/Newsletter/April 2025/Contents/Memory of the World report|Memory of the World report]]: Surging forward in Spanish and Arabic
* [[outreach:GLAM/Newsletter/April 2025/Contents/Events|Calendar]]: May's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/April 2025/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 20:53, 11 May 2025 (UTC)&lt;/div&gt;
&lt;!-- Message sent by User:Romaine@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/GLAM&amp;oldid=28668932 --&gt;

== ''This Month in GLAM'': May 2025 ==

{| style=&quot;width:100%;&quot;
| valign=&quot;top&quot; align=&quot;center&quot; style=&quot;border:1px gray solid; padding:1em;&quot; |
{| align=&quot;center&quot;
|-
| style=&quot;text-align: center;&quot; | [[File:This Month in GLAM logo 2018.png|350px|center|link=outreach:GLAM/Newsletter]]&lt;br /&gt;
&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/May 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue V, May 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
| &lt;span style=&quot;font-size:12pt; font-family:Times New Roman;&quot;&gt; '''&lt;u&gt;Headlines&lt;/u&gt;'''&lt;/span&gt;
|- style=&quot;font-size:10pt; font-family:Times New Roman; text-align:center;&quot;
| &lt;div style=&quot;text-align:left; column-count:2; column-width:28em; vertical-align:top;&quot;&gt;

* [[outreach:GLAM/Newsletter/May 2025/Contents/Albania report|Albania report]]: Enhancing the LGBTQ+ content in Albanian Wikipedia
* [[outreach:GLAM/Newsletter/May 2025/Contents/Belgium report|Belgium report]]: Belgium Public Domain Day and Dance Heritage
* [[outreach:GLAM/Newsletter/May 2025/Contents/Brazil report|Brazil report]]: Video resource on Wikimedia Brasil and Casa de Oswaldo Cruz Partnership Released
* [[outreach:GLAM/Newsletter/May 2025/Contents/Croatia report|Croatia report]]: SPRINGing back activities
* [[outreach:GLAM/Newsletter/May 2025/Contents/Czech Republic report|Czech Republic report]]: National Library of the CR events and important guests
* [[outreach:GLAM/Newsletter/May 2025/Contents/Europe report|Europe report]]: DARIAH DHwiki WG coming activity 
* [[outreach:GLAM/Newsletter/May 2025/Contents/India report|India report]]: GLAM project starts with Nanda Talukdar Foundation at Jorhat, Assam
* [[outreach:GLAM/Newsletter/May 2025/Contents/Indonesia report|Indonesia report]]: GLAM Wiki Month 2025 in Indonesia
* [[outreach:GLAM/Newsletter/May 2025/Contents/Italy report|Italy report]]: From charts to concrete
* [[outreach:GLAM/Newsletter/May 2025/Contents/Kosovo report|Kosovo report]]: Traditional Albanian Food Photography Competition 2025
* [[outreach:GLAM/Newsletter/May 2025/Contents/New Zealand report|New Zealand report]]: Update from Auckland Museum; Let the Wikifying Commence; Listful Thinking 
* [[outreach:GLAM/Newsletter/May 2025/Contents/Nigeria report|Nigeria report]]: Architectural Folklore Campaign Series
* [[outreach:GLAM/Newsletter/May 2025/Contents/Poland report|Poland report]]: What's up in GLAM in Poland
* [[outreach:GLAM/Newsletter/May 2025/Contents/Serbia report|Serbia report]]: Celebrating Museums and strengthening #1Lib1Ref connections
* [[outreach:GLAM/Newsletter/May 2025/Contents/Spain report|Spain report]]: Some news from Spain
* [[outreach:GLAM/Newsletter/May 2025/Contents/Switzerland report|Switzerland report]]: International Museum Day 2025; CoCreation PTT-Archive; Scoring Girls
* [[outreach:GLAM/Newsletter/May 2025/Contents/UK report|UK report]]: The 18th language
* [[outreach:GLAM/Newsletter/May 2025/Contents/Ukraine report|Ukraine report]]: Spring GLAM news from Ukraine – first major survey for GLAM institutions &amp; yet another successful #1Lib1Ref
* [[outreach:GLAM/Newsletter/May 2025/Contents/USA report|USA report]]: May meetings
* [[outreach:GLAM/Newsletter/May 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: BHL-Wiki Working Group May monthly highlights
* [[outreach:GLAM/Newsletter/May 2025/Contents/Memory of the World report|Memory of the World report]]: Preparing the data upload
* [[outreach:GLAM/Newsletter/May 2025/Contents/Events|Calendar]]: June's GLAM events

&lt;/div&gt;
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| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/May 2025/Single|Single-page]]
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To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 06:00, 10 June 2025 (UTC)&lt;/div&gt;
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== ''This Month in GLAM'': June 2025 ==

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&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/June 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue VI, June 2025&lt;/span&gt;]]&lt;/div&gt;
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|- style=&quot;text-align: center;&quot;
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* [[outreach:GLAM/Newsletter/June 2025/Contents/Australia report|Australia report]]: Celebrating Communication History and Women Artists through Wikipedia
* [[outreach:GLAM/Newsletter/June 2025/Contents/Belgium report|Belgium report]]: Public Domain Day in Europe 2026
* [[outreach:GLAM/Newsletter/June 2025/Contents/Brazil report|Brazil report]]: Expanding Data on Maranhão's Heritage
* [[outreach:GLAM/Newsletter/June 2025/Contents/Czech Republic report|Czech Republic report]]: First call of Science Month at Wikipedia in the Czech Republic wrapped-up
* [[outreach:GLAM/Newsletter/June 2025/Contents/France report|France report]]: WiR and science&amp;GLAM tour in France
* [[outreach:GLAM/Newsletter/June 2025/Contents/Indonesia report|Indonesia report]]: Another publication from Grant for GLAM Indonesia program &amp; Minangkabau Wikisource Competition upadates 
* [[outreach:GLAM/Newsletter/June 2025/Contents/Italy report|Italy report]]: Isoseismals &amp; Icons
* [[outreach:GLAM/Newsletter/June 2025/Contents/Kosovo report|Kosovo report]]: Celebrating Albanian Cuisine Through Photography: Winners of the 2025 Contest Announced!
* [[outreach:GLAM/Newsletter/June 2025/Contents/Mexico report|Mexico report]]: Open Cultural Data Hackathon in Puebla
* [[outreach:GLAM/Newsletter/June 2025/Contents/New Zealand report|New Zealand report]]: Wikifying the International Congress of History of Science and Technology; Librarians and Wikipedia; NZ species edit-a-thons
* [[outreach:GLAM/Newsletter/June 2025/Contents/Nigeria report|Nigeria report]]: Mapping Heritage Buildings on Wikidata and Wiki Heritage Fellowship
* [[outreach:GLAM/Newsletter/June 2025/Contents/Macedonia report|North Macedonia report]]: Wikimedia MKD Strengthens Ties with Academic Institutions: A Wikimedian-in-Residence at the Institute of Macedonian Literature 
* [[outreach:GLAM/Newsletter/June 2025/Contents/Poland report|Poland report]]: What's up in GLAM in Poland
* [[outreach:GLAM/Newsletter/June 2025/Contents/Portugal report|Portugal report]]: Registration is now open for GLAM WikiCon 2025!
* [[outreach:GLAM/Newsletter/June 2025/Contents/Serbia report|Serbia report]]: June highlights from Wikimedia Serbia and beginning of new accredited seminar
* [[outreach:GLAM/Newsletter/June 2025/Contents/Switzerland report|Switzerland report]]: International Archives Week 2025; SAPA Performing Arts; GLAM Meeting Biel/Bienne
* [[outreach:GLAM/Newsletter/June 2025/Contents/UK report|UK report]]: New Featured Pictures
* [[outreach:GLAM/Newsletter/June 2025/Contents/USA report|USA report]]: June meetings
* [[outreach:GLAM/Newsletter/June 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: BHL Transition update, LivingData2025 and Women Genera paper
* [[outreach:GLAM/Newsletter/June 2025/Contents/Memory of the World report|Memory of the World report]]: Data upload achieved
* [[outreach:GLAM/Newsletter/June 2025/Contents/Wiki Knowledge Park report|Wiki Loves Ramadan 2025 report]]: Celebrating Heritage and Faith: Wiki Loves Ramadan 2025
* [[outreach:GLAM/Newsletter/June 2025/Contents/Events|Calendar]]: July's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/June 2025/Single|Single-page]]
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| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 15:34, 12 July 2025 (UTC)&lt;/div&gt;
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== ''This Month in GLAM'': July 2025 ==

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&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/July 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue VII, July 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
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* [[outreach:GLAM/Newsletter/July 2025/Contents/Albania report|Albania report]]: 10 years of Wikimedians of Albanian Language User Group
* [[outreach:GLAM/Newsletter/July 2025/Contents/Aruba report|Aruba report]]: Wikipedia on Aruba – From Island to Archive: Aruba’s Journey on Wikimedia
* [[outreach:GLAM/Newsletter/July 2025/Contents/Belgium report|Belgium report]]: De Standaard Solidarity Prize and How to Upload Artwork
* [[outreach:GLAM/Newsletter/July 2025/Contents/New Zealand report|New Zealand report]]: Women in Wartime
* [[outreach:GLAM/Newsletter/July 2025/Contents/Nigeria report|Nigeria report]]: Expanding Access to Heritage Knowledge
* [[outreach:GLAM/Newsletter/July 2025/Contents/Portugal report|Portugal report]]: GLAM Wiki 2025: Program Highlights &amp; Volunteer Call
* [[outreach:GLAM/Newsletter/July 2025/Contents/Serbia report|Serbia report]]: July in Wikimedia Serbia
* [[outreach:GLAM/Newsletter/July 2025/Contents/Switzerland report|Switzerland report]]: Poster Exhibition, Screening Public Viewing, Women Memorials 
* [[outreach:GLAM/Newsletter/July 2025/Contents/UK report|UK report]]: Japanese art in Arabic
* [[outreach:GLAM/Newsletter/July 2025/Contents/USA report|USA report]]: Wikinics &amp; Edit-a-thons
* [[outreach:GLAM/Newsletter/July 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: BHL blog by Tiago Lubiana, BHL Wikimedian in Residence, and news on the BHL transition
* [[outreach:GLAM/Newsletter/July 2025/Contents/AvoinGLAM report|AvoinGLAM report]]: Introducing Oulu Löyly – we want to hear your thoughts!
* [[outreach:GLAM/Newsletter/July 2025/Contents/Memory of the World report|Memory of the World report]]: Getting the message out
* [[outreach:GLAM/Newsletter/July 2025/Contents/WMF GLAM report|WMF GLAM report]]: Culture &amp; Heritage team transitioning to broader Content Enablement team
* [[outreach:GLAM/Newsletter/July 2025/Contents/Events|Calendar]]: August's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/July 2025/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 19:58, 11 August 2025 (UTC)&lt;/div&gt;
&lt;!-- Message sent by User:Romaine@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/GLAM&amp;oldid=29083949 --&gt;

== ''This Month in GLAM'': August 2025 ==

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| style=&quot;text-align: center;&quot; | [[File:This Month in GLAM logo 2018.png|350px|center|link=outreach:GLAM/Newsletter]]&lt;br /&gt;
&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/August 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue VIII, August 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
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* [[outreach:GLAM/Newsletter/August 2025/Contents/Albania report|Albania report]]: Wikipedia Pages Wanting Photos campaign 2025 in Albania and Kosovo 
* [[outreach:GLAM/Newsletter/August 2025/Contents/Brazil report|Brazil report]]: Expanding Cultural Heritage in Brazil: School communities, Wikisource course, GLAM-Wiki Impact and Wiki Takes Alcântara
* [[outreach:GLAM/Newsletter/August 2025/Contents/India report|India report]]: Digitization starts at two more libraries in West Bengal
* [[outreach:GLAM/Newsletter/August 2025/Contents/Indonesia report|Indonesia report]]: Grant for GLAM Indonesia is open!
* [[outreach:GLAM/Newsletter/August 2025/Contents/Italy report|Italy report]]: From food festivals to PhD courses: Wikimedia in Italian academia
* [[outreach:GLAM/Newsletter/August 2025/Contents/New Zealand report|New Zealand report]]: NZ species edit-a-thons, a scholarly article, a course, and Auckland Museum editors at Wikimania
* [[outreach:GLAM/Newsletter/August 2025/Contents/Nigeria report|Nigeria report]]: Highlights from Wiki Heritage Fellowship and Policy Advocacy in Nigeria
* [[outreach:GLAM/Newsletter/August 2025/Contents/Macedonia report|North Macedonia report]]: Wiki Loves Film – Collaboration with MakeDox Film Festival
* [[outreach:GLAM/Newsletter/August 2025/Contents/Poland report|Poland report]]: WikiChełmoński: When Wikipedia Becomes a Guide in the Museum
* [[outreach:GLAM/Newsletter/August 2025/Contents/Switzerland report|Switzerland report]]: BAM Hackathon, 3D for Cultural Heritage, Wiki Cite 
* [[outreach:GLAM/Newsletter/August 2025/Contents/UK report|UK report]]: Working towards more image sharing
* [[outreach:GLAM/Newsletter/August 2025/Contents/USA report|USA report]]: Wiknics
* [[outreach:GLAM/Newsletter/August 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: News on BHL transition and updates on work by BHLWiki Working Group members
* [[outreach:GLAM/Newsletter/August 2025/Contents/AvoinGLAM report|AvoinGLAM report]]: Steps towards a sustainable cultural commons
* [[outreach:GLAM/Newsletter/August 2025/Contents/Memory of the World report|Memory of the World report]]: A pivotal month
* [[outreach:GLAM/Newsletter/August 2025/Contents/Wikisource report|Wikisource report]]: Wikisource Reader app released on Google Play Store
* [[outreach:GLAM/Newsletter/August 2025/Contents/Events|Calendar]]: September's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/August 2025/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 18:24, 11 September 2025 (UTC)&lt;/div&gt;
&lt;!-- Message sent by User:Romaine@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/GLAM&amp;oldid=29253474 --&gt;

== ''This Month in GLAM'': September 2025 ==

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&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/September 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue IX, September 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
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* [[outreach:GLAM/Newsletter/September 2025/Contents/Albania report|Albania report]]: Wikipedia edit-a-thon at the Skampa Theater in Elbasan, Albania 
* [[outreach:GLAM/Newsletter/September 2025/Contents/Aruba report|Aruba report]]: Wikipedia on Aruba Initiative: Building Knowledge and Visibility 
* [[outreach:GLAM/Newsletter/September 2025/Contents/Australia report|Australia report]]: State Library Victoria WikiFest
* [[outreach:GLAM/Newsletter/September 2025/Contents/Belgium report|Belgium report]]: Make cultural heritage freely accessible, Hasselt's collections shine online &amp; Let's give women a voice on Wikipedia!
* [[outreach:GLAM/Newsletter/September 2025/Contents/Italy report|Italy report]]: Hidden heritage unveiled: science, history and nature on Wikimedia
* [[outreach:GLAM/Newsletter/September 2025/Contents/Netherlands report|Netherlands report]]: Wikimedia Commons birthday content donation
* [[outreach:GLAM/Newsletter/September 2025/Contents/New Zealand report|New Zealand report]]: Wikimedian in Residence at the Bioeconomy Science Institute &amp; Wikiproject NZ Women in Architecture
* [[outreach:GLAM/Newsletter/September 2025/Contents/Macedonia report|North Macedonia report]]: Expanding Access to Culture and Knowledge through Key Partnerships
* [[outreach:GLAM/Newsletter/September 2025/Contents/Poland report|Poland report]]: WikiChełmoński: the Special Guided Tour at the National Museum in Krakow
* [[outreach:GLAM/Newsletter/September 2025/Contents/Portugal report|Portugal report]]: GLAM Wiki 2025: Join the Conference Online and Explore Lisbon’s Culture
* [[outreach:GLAM/Newsletter/September 2025/Contents/Switzerland report|Switzerland report]]: LibreABC, Feminist Voices, Grade Conference
* [[outreach:GLAM/Newsletter/September 2025/Contents/UK report|UK report]]: Awards season
* [[outreach:GLAM/Newsletter/September 2025/Contents/USA report|USA report]]: September edit-a-thons &amp; meetings
* [[outreach:GLAM/Newsletter/September 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: Updates on work by BHLWiki Working Group members
* [[outreach:GLAM/Newsletter/September 2025/Contents/Memory of the World report|Memory of the World report]]: The Memory of the World wiki challenge
* [[outreach:GLAM/Newsletter/September 2025/Contents/Events|Calendar]]: October's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/September 2025/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 12:45, 9 October 2025 (UTC)&lt;/div&gt;
&lt;!-- Message sent by User:Romaine@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/GLAM&amp;oldid=29409147 --&gt;

== ''This Month in GLAM'': October 2025 ==

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&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/October 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue X, October 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
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* [[outreach:GLAM/Newsletter/October 2025/Contents/Belgium report|Belgium report]]: From a small language to Wikipedia's Biggest, Share Your Story with the Industry Museum and War Diaries
* [[outreach:GLAM/Newsletter/October 2025/Contents/Croatia report|Croatia report]]: Autumn activities
* [[outreach:GLAM/Newsletter/October 2025/Contents/Indonesia report|Indonesia report]]: WikiCommon community meet-ups, Wikisource trainings, and the announcement of Grant for GLAM Indonesia
* [[outreach:GLAM/Newsletter/October 2025/Contents/Italy report|Italy report]]: Open culture on stage
* [[outreach:GLAM/Newsletter/October 2025/Contents/Mexico report|Mexico report]]: How Wikimedia México is training cultural and government institutions as wikimedians.
* [[outreach:GLAM/Newsletter/October 2025/Contents/Netherlands report|Netherlands report]]: Open Topstukken project concludes; Network Archives Design and Digital Culture
* [[outreach:GLAM/Newsletter/October 2025/Contents/New Zealand report|New Zealand report]]: Granny's Wonderful Chair, Preparing for Auckland Museum Wiki Summer Students and an the ASBS Introductory Wiki Webinar
* [[outreach:GLAM/Newsletter/October 2025/Contents/Nigeria report|Nigeria report]]: Report on Participation at the 12th International Youth Conference
* [[outreach:GLAM/Newsletter/October 2025/Contents/Poland report|Poland report]]: Promoting Open Data and Digital Commons in Culture and Research
* [[outreach:GLAM/Newsletter/October 2025/Contents/Portugal report|Portugal report]]: GLAM Wiki Conference 2025 Wrap-Up
* [[outreach:GLAM/Newsletter/October 2025/Contents/Serbia report|Serbia report]]: October in Wikimedia Serbia
* [[outreach:GLAM/Newsletter/October 2025/Contents/Switzerland report|Switzerland report]]: DaSCHcon, 3D, Wiki GLAM conference
* [[outreach:GLAM/Newsletter/October 2025/Contents/UK report|UK report]]: A look at Grokipedia
* [[outreach:GLAM/Newsletter/October 2025/Contents/USA report|USA report]]: October edit-a-thons &amp; meetings
* [[outreach:GLAM/Newsletter/October 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: LivingData 2025, GLAMWiki and a &quot;collector&quot; Wikidata property proposal
* [[outreach:GLAM/Newsletter/October 2025/Contents/Memory of the World report|Memory of the World report]]: International outreach
* [[outreach:GLAM/Newsletter/October 2025/Contents/Sustainable CultureConnect Project report|Sustainable CultureConnect Project report]]: Sustainable CultureConnect: Empowering Youth and Preserving Heritage through Open Knowledge and Leadership
* [[outreach:GLAM/Newsletter/October 2025/Contents/Events|Calendar]]: November's GLAM events

&lt;/div&gt;
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| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/October 2025/Single|Single-page]]
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| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 11:28, 10 November 2025 (UTC)&lt;/div&gt;
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== ''This Month in GLAM'': November 2025 ==

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&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/November 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue XI, November 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
| &lt;span style=&quot;font-size:12pt; font-family:Times New Roman;&quot;&gt; '''&lt;u&gt;Headlines&lt;/u&gt;'''&lt;/span&gt;
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* [[outreach:GLAM/Newsletter/November 2025/Contents/Belgium report|Belgium report]]: Call for volunteers, Computer's Day and upcoming events
* [[outreach:GLAM/Newsletter/November 2025/Contents/Czech Republic report|Czech Republic report]]: Czech Radio and Wikimedia CZ launched cooperation
* [[outreach:GLAM/Newsletter/November 2025/Contents/Indonesia report|Indonesia report]]: Our Activities in November
* [[outreach:GLAM/Newsletter/November 2025/Contents/Italy report|Italy report]]: Wrap up of the 2025 GLAM call
* [[outreach:GLAM/Newsletter/November 2025/Contents/New Zealand report|New Zealand report]]: ASBS 2025 Conference, a NZBSI Wikimedian in Residence update &amp; Auckland Museum Summer Students 
* [[outreach:GLAM/Newsletter/November 2025/Contents/Macedonia report|North Macedonia report]]: Wikimedia MKD and Cultural Institutions: A Year of Growth, Content, and Collaboration
* [[outreach:GLAM/Newsletter/November 2025/Contents/Poland report|Poland report]]: GLAM Wiki 2025 conference, WiR Meeting and the First National Institute of Museums Training on GLAM–Wiki
* [[outreach:GLAM/Newsletter/November 2025/Contents/Switzerland report|Switzerland report]]: IT Wikicon, Faces and Masks, Matrimoine @ Genève
* [[outreach:GLAM/Newsletter/November 2025/Contents/UK report|UK report]]: Awards season- again!
* [[outreach:GLAM/Newsletter/November 2025/Contents/USA report|USA report]]: November edit-a-thons &amp; meetings
* [[outreach:GLAM/Newsletter/November 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: Updates on work by BHLWiki Working Group members
* [[outreach:GLAM/Newsletter/November 2025/Contents/Content Partnerships Hub report|Content Partnerships Hub report]]: Here to Help: The Next Stage of the Content Partnerships Hub
* [[outreach:GLAM/Newsletter/November 2025/Contents/Memory of the World report|Memory of the World report]]: Enriching the data set
* [[outreach:GLAM/Newsletter/November 2025/Contents/Events|Calendar]]: December's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/November 2025/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 04:21, 11 December 2025 (UTC)&lt;/div&gt;
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== ''This Month in GLAM'': December 2025 ==

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&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/December 2025|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XV, Issue XII, December 2025&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
| &lt;span style=&quot;font-size:12pt; font-family:Times New Roman;&quot;&gt; '''&lt;u&gt;Headlines&lt;/u&gt;'''&lt;/span&gt;
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* [[outreach:GLAM/Newsletter/December 2025/Contents/From the team|From the team]]: Global GLAM Calls Continue in 2026
* [[outreach:GLAM/Newsletter/December 2025/Contents/Albania report|Albania report]]: Wikimedian in Residence, 2025, in Elbasan Albania
* [[outreach:GLAM/Newsletter/December 2025/Contents/Argentina report|Argentina report]]: Resume of the year
* [[outreach:GLAM/Newsletter/December 2025/Contents/Australia report|Australia report]]: AMaGA partnership, signing the Open Heritage Statement and South Australian Museum Partner Project 
* [[outreach:GLAM/Newsletter/December 2025/Contents/Belgium report|Belgium report]]: Public Domain Day 2026, Circus Heritage and Fosdem
* [[outreach:GLAM/Newsletter/December 2025/Contents/Colombia report|Colombia report]]: Preparándonos para celebrar el día del dominio público - Getting ready for the Public Domain Day celebration
* [[outreach:GLAM/Newsletter/December 2025/Contents/Italy report|Italy report]]: Exploring Italy and Unlocking Its Heritage: Touring Club Italiano and GLAM Call 2026–2028
* [[outreach:GLAM/Newsletter/December 2025/Contents/New Zealand report|New Zealand report]]: ASBS 2025 Conference follow-up Wiki webinar &amp; Auckland Museum student update
* [[outreach:GLAM/Newsletter/December 2025/Contents/Poland report|Poland report]]: Public Domain, Conferences, and Conversations on Open Culture
* [[outreach:GLAM/Newsletter/December 2025/Contents/Switzerland report|Switzerland report]]: GLAM on Tour Bellinzona, Xmas Event, GLAM Wiki Group
* [[outreach:GLAM/Newsletter/December 2025/Contents/UK report|UK report]]: 2025 in review
* [[outreach:GLAM/Newsletter/December 2025/Contents/USA report|USA report]]: December meetings
* [[outreach:GLAM/Newsletter/December 2025/Contents/Public Domain Day report|Public Domain Day report]]: Public Domain Day 2026
* [[outreach:GLAM/Newsletter/December 2025/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: Updates on work by BHLWiki Working Group members
* [[outreach:GLAM/Newsletter/December 2025/Contents/Memory of the World report|Memory of the World report]]: 2025 in review
* [[outreach:GLAM/Newsletter/December 2025/Contents/Events|Calendar]]: January's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/December 2025/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 12:57, 12 January 2026 (UTC)&lt;/div&gt;
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== ''This Month in GLAM'': January 2026 ==

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| style=&quot;text-align: center;&quot; | [[File:This Month in GLAM logo 2018.png|350px|center|link=outreach:GLAM/Newsletter]]&lt;br /&gt;
&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/January 2026|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XVI, Issue I, January 2026&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
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* [[outreach:GLAM/Newsletter/January 2026/Contents/From the team|From the team]]: Wikipedia at 25
* [[outreach:GLAM/Newsletter/January 2026/Contents/Aruba report|Aruba report]]: Pap-Wikipedia Turns 20: A Milestone for Papiamento/u Knowledge
* [[outreach:GLAM/Newsletter/January 2026/Contents/Colombia report|Colombia report]]: Celebrando el día del dominio público 2026/Celebrating the public domain day 2026 
* [[outreach:GLAM/Newsletter/January 2026/Contents/France report|France report]]: Wikipedia's Birthday in France
* [[outreach:GLAM/Newsletter/January 2026/Contents/Germany report|Germany report]]: Gifted by our friends: Loads of presents for Wikipedia25 by German and Austrian GLAMs
* [[outreach:GLAM/Newsletter/January 2026/Contents/Indonesia report|Indonesia report]]: Activities in December-January
* [[outreach:GLAM/Newsletter/January 2026/Contents/Italy report|Italy report]]: Knowledge in action: Barindelli collection and Wikipedia 25
* [[outreach:GLAM/Newsletter/January 2026/Contents/Netherlands report|Netherlands report]]: Image donation Dutch Book History
* [[outreach:GLAM/Newsletter/January 2026/Contents/New Zealand report|New Zealand report]]: Auckland Museum Student Edit-a-thon, the NZBSI Wikimedian in Residence project, and other residencies
* [[outreach:GLAM/Newsletter/January 2026/Contents/Macedonia report|North Macedonia report]]: GLAM Program of Wikimedia MKD – 2026 Overview
* [[outreach:GLAM/Newsletter/January 2026/Contents/Poland report|Poland report]]: Explore Historic Portraits from the Museum of Photography in Kraków
* [[outreach:GLAM/Newsletter/January 2026/Contents/Serbia report|Serbia report]]: January in Wikimedia Serbia
* [[outreach:GLAM/Newsletter/January 2026/Contents/Switzerland report|Switzerland report]]: Museum in Chiasso, Alpine Museum, Women Monuments
* [[outreach:GLAM/Newsletter/January 2026/Contents/UK report|UK report]]: Sharing more of the Enamels of the World
* [[outreach:GLAM/Newsletter/January 2026/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: Updates on work by BHLWiki Working Group members
* [[outreach:GLAM/Newsletter/January 2026/Contents/Map the GLAM report|Map the GLAM report]]: Wiki and GLAM: Harnessing Knowledge to Foster Gender Equality
* [[outreach:GLAM/Newsletter/January 2026/Contents/Memory of the World report|Memory of the World report]]: Manuscripts on Arabic Wikipedia
* [[outreach:GLAM/Newsletter/January 2026/Contents/Wikidata report|Wikidata report]]: Two key Wikidata Requests for Comments, relevant for future GLAM-Wiki work
* [[outreach:GLAM/Newsletter/January 2026/Contents/Events|Calendar]]: February's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/January 2026/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 08:34, 11 February 2026 (UTC)&lt;/div&gt;
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== ''This Month in GLAM'': February 2026 ==

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| style=&quot;text-align: center;&quot; | [[File:This Month in GLAM logo 2018.png|350px|center|link=outreach:GLAM/Newsletter]]&lt;br /&gt;
&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/February 2026|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XVI, Issue II, February 2026&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
| &lt;span style=&quot;font-size:12pt; font-family:Times New Roman;&quot;&gt; '''&lt;u&gt;Headlines&lt;/u&gt;'''&lt;/span&gt;
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* [[outreach:GLAM/Newsletter/February 2026/Contents/From the team|From the team]]: GLAM in 2026 is calling to action!
* [[outreach:GLAM/Newsletter/February 2026/Contents/Belgium report|Belgium report]]: A photo safari through Wiki Loves Fashion &amp; a rescue mission with Mission Gourmande!
* [[outreach:GLAM/Newsletter/February 2026/Contents/Czech Republic report|Czech Republic report]]: National Library overview of 2025
* [[outreach:GLAM/Newsletter/February 2026/Contents/Italy report|Italy report]]: Cultural Heritage and Memory: 2026 GLAM Call and Executed Renaissance Edit-a-thon
* [[outreach:GLAM/Newsletter/February 2026/Contents/New Zealand report|New Zealand report]]: Farewelling the Auckland Museum Summer Students and an update on the NZBSI WiR
* [[outreach:GLAM/Newsletter/February 2026/Contents/Poland report|Poland report]]: Wikipedia in Cultural Marketing at Crash Mondays Warsaw
* [[outreach:GLAM/Newsletter/February 2026/Contents/Serbia report|Serbia report]]: February in Wikimedia Serbia
* [[outreach:GLAM/Newsletter/February 2026/Contents/Spain report|Spain report]]: Wikidata in the GLAM context II
* [[outreach:GLAM/Newsletter/February 2026/Contents/Switzerland report|Switzerland report]]: Neocomensia, SAPA, Atelier Winterthur, GLAM-on-Tour Disentis
* [[outreach:GLAM/Newsletter/February 2026/Contents/UK report|UK report]]: New content in African and Asian languages
* [[outreach:GLAM/Newsletter/February 2026/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: Updates on work by BHLWiki Working Group members
* [[outreach:GLAM/Newsletter/February 2026/Contents/Memory of the World report|Memory of the World report]]: Focus on Indigenous issues
* [[outreach:GLAM/Newsletter/February 2026/Contents/Events|Calendar]]: March's GLAM events
&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/February 2026/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 00:52, 12 March 2026 (UTC)&lt;/div&gt;
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== ''This Month in GLAM'': March 2026 ==

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&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/March 2026|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XVI, Issue III, March 2026&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
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* [[outreach:GLAM/Newsletter/March 2026/Contents/From the team|From the team]]: GLAM in Equinox
* [[outreach:GLAM/Newsletter/March 2026/Contents/Albania report|Albania report]]: Wikigap 2026 in Tirana, Albania
* [[outreach:GLAM/Newsletter/March 2026/Contents/Aruba report|Aruba report]]: Celebrating 20 Years of Papiamentu/o Wikipedia
* [[outreach:GLAM/Newsletter/March 2026/Contents/Bolivia report|Bolivia report]]: The Family Treasures Project Returns, with New GLAM Partners
* [[outreach:GLAM/Newsletter/March 2026/Contents/Brazil report|Brazil report]]: Wiki Loves Folklore Brazil defines six categories to represent Brazilian culture on Commons 
* [[outreach:GLAM/Newsletter/March 2026/Contents/Colombia report|Colombia report]]: Architecture in the public domain in libraries in Nariño/Arquitectura en dominio público en bibliotecas nariñenses
* [[outreach:GLAM/Newsletter/March 2026/Contents/Czech Republic report|Czech Republic report]]: Main Wikimedian in Residence report for 2025 is here
* [[outreach:GLAM/Newsletter/March 2026/Contents/Germany report|Germany report]]: Art History Loves Wiki 2026 – digital/local. collection loves wiki
* [[outreach:GLAM/Newsletter/March 2026/Contents/India report|India report]]: Collaboration resumes with the British Library for Bangla Wikisource
* [[outreach:GLAM/Newsletter/March 2026/Contents/Italy report|Italy report]]: 2026 winners projects
* [[outreach:GLAM/Newsletter/March 2026/Contents/New Zealand report|New Zealand report]]: New Zealand Bioeconomy Science Institute Wikimedian in Residence update, a letter published in Nature &amp; Architecture + Women NZ
* [[outreach:GLAM/Newsletter/March 2026/Contents/Macedonia report|North Macedonia report]]: Wikimedia MKD's GLAM Highlights: Botany and Beyond 
* [[outreach:GLAM/Newsletter/March 2026/Contents/Portugal report|Portugal report]]: March in Portugal
* [[outreach:GLAM/Newsletter/March 2026/Contents/Serbia report|Serbia report]]: March in Wikimedia Serbia
* [[outreach:GLAM/Newsletter/March 2026/Contents/Sweden report|Sweden report]]: Looking for similar images
* [[outreach:GLAM/Newsletter/March 2026/Contents/Switzerland report|Switzerland report]]: GLAM Wiki Group, Donna, CoCreation PTT-Archive
* [[outreach:GLAM/Newsletter/March 2026/Contents/UK report|UK report]]: CILIP MDG 2026 Conference, Library meetups and the World's Enamels
* [[outreach:GLAM/Newsletter/March 2026/Contents/Ukraine report|Ukraine report]]: Spring 2026 news from Ukraine – Wiki Loves Folklore &amp; more #1Lib1Ref
* [[outreach:GLAM/Newsletter/March 2026/Contents/USA report|USA report]]: MoMA's International Traveling Exhibitions and Wikidata on the LD4 Arts Affinity Group April Community Call 
* [[outreach:GLAM/Newsletter/March 2026/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: Updates on work by BHL-Wiki Working Group members
* [[outreach:GLAM/Newsletter/March 2026/Contents/AvoinGLAM report|AvoinGLAM report]]: Oulu Löyly
* [[outreach:GLAM/Newsletter/March 2026/Contents/Memory of the World report|Memory of the World report]]: Indigenous languages on the Main Page
* [[outreach:GLAM/Newsletter/March 2026/Contents/Events|Calendar]]: April's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/March 2026/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 18:19, 9 April 2026 (UTC)&lt;/div&gt;
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== ''This Month in GLAM'': April 2026 ==

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&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/April 2026|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XVI, Issue IV, April 2026&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
| &lt;span style=&quot;font-size:12pt; font-family:Times New Roman;&quot;&gt; '''&lt;u&gt;Headlines&lt;/u&gt;'''&lt;/span&gt;
|- style=&quot;font-size:10pt; font-family:Times New Roman; text-align:center;&quot;
| &lt;div style=&quot;text-align:left; column-count:2; column-width:28em; vertical-align:top;&quot;&gt;

* [[outreach:GLAM/Newsletter/April 2026/Contents/From the team|From the team]]: How the GLAM Community Can Shine in the 2026–27 Annual Plan
* [[outreach:GLAM/Newsletter/April 2026/Contents/Albania report|Albania report]]: International Roma Day 2026 in Tirana, Albania
* [[outreach:GLAM/Newsletter/April 2026/Contents/Argentina report|Argentina report]]: WikiConf Argentina and GLAM projects
* [[outreach:GLAM/Newsletter/April 2026/Contents/Asia report|Asia report]]: Documenting and citing oral knowledge in audio and video
* [[outreach:GLAM/Newsletter/April 2026/Contents/Australia report|Australia report]]: WikiCon Australia, ICIP, Orphan works and Trans-Tasman partnerships
* [[outreach:GLAM/Newsletter/April 2026/Contents/Brazil report|Brazil report]]: Wikimedia Brasil publishes book on the power and challenges of free knowledge
* [[outreach:GLAM/Newsletter/April 2026/Contents/Colombia report|Colombia report]]: We celebrate Public Domain Day with an expert panel / Celebramos el día del dominio público con un Panel de expertas
* [[outreach:GLAM/Newsletter/April 2026/Contents/Italy report|Italy report]]: Ongoing and New GLAM-Wiki Projects
* [[outreach:GLAM/Newsletter/April 2026/Contents/New Zealand report|New Zealand report]]: Women in Wartime event at Auckland Museum &amp; an update for WiR NZBSI
* [[outreach:GLAM/Newsletter/April 2026/Contents/Nigeria report|Nigeria report]]: Wikimedia Commons Upload Campaign
* [[outreach:GLAM/Newsletter/April 2026/Contents/Macedonia report|North Macedonia report]]: Wikimedia MKD in Action: Digitization, Wikisource and Educational Workshops
* [[outreach:GLAM/Newsletter/April 2026/Contents/Poland report|Poland report]]: GLAM-Wiki Developments: Residencies, Partnerships and Audiovisual Heritage
* [[outreach:GLAM/Newsletter/April 2026/Contents/Serbia report|Serbia report]]: April in Wikimedia Serbia
* [[outreach:GLAM/Newsletter/April 2026/Contents/Switzerland report|Switzerland report]]: Le Donne di Villa Massimo
* [[outreach:GLAM/Newsletter/April 2026/Contents/UK report|UK report]]: A tenth-century Quran and Islamic Art in Urdu
* [[outreach:GLAM/Newsletter/April 2026/Contents/USA report|USA report]]: Wiki MIT launches and US meetups
* [[outreach:GLAM/Newsletter/April 2026/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: Update on the BHL Annual Meeting 27 April - 1 May &amp; BHL Day
* [[outreach:GLAM/Newsletter/April 2026/Contents/Memory of the World report|Memory of the World report]]: Eight new articles on MoW inscriptions
* [[outreach:GLAM/Newsletter/April 2026/Contents/Events|Calendar]]: May's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/April 2026/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 13:50, 11 May 2026 (UTC)&lt;/div&gt;
&lt;!-- Message sent by User:Romaine@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/GLAM&amp;oldid=30513880 --&gt;

== ''This Month in GLAM'': May 2026 ==

{| style=&quot;width:100%;&quot;
| valign=&quot;top&quot; align=&quot;center&quot; style=&quot;border:1px gray solid; padding:1em;&quot; |
{| align=&quot;center&quot;
|-
| style=&quot;text-align: center;&quot; | [[File:This Month in GLAM logo 2018.png|350px|center|link=outreach:GLAM/Newsletter]]&lt;br /&gt;
&lt;hr /&gt;
&lt;div style=&quot;font-size:12pt; font-family:Times New Roman; text-align:center;&quot;&gt;[[outreach:GLAM/Newsletter/May 2026|&lt;span style=&quot;color:darkslategray;&quot;&gt;This Month in GLAM – Volume XVI, Issue V, May 2026&lt;/span&gt;]]&lt;/div&gt;
&lt;hr /&gt;&lt;br /&gt;

|- style=&quot;text-align: center;&quot;
| &lt;span style=&quot;font-size:12pt; font-family:Times New Roman;&quot;&gt; '''&lt;u&gt;Headlines&lt;/u&gt;'''&lt;/span&gt;
|- style=&quot;font-size:10pt; font-family:Times New Roman; text-align:center;&quot;
| &lt;div style=&quot;text-align:left; column-count:2; column-width:28em; vertical-align:top;&quot;&gt;

* [[outreach:GLAM/Newsletter/May 2026/Contents/From the team|From the team]]: May Global GLAM Call Recording and June Call Announcement
* [[outreach:GLAM/Newsletter/May 2026/Contents/Australia report|Australia report]]: Celebrating #1Lib1Ref Australasia
* [[outreach:GLAM/Newsletter/May 2026/Contents/Brazil report|Brazil report]]: GLAM Connection: Fashion, History, and Biodiversity in May
* [[outreach:GLAM/Newsletter/May 2026/Contents/France report|France report]]: New Wikimedian in Residence
* [[outreach:GLAM/Newsletter/May 2026/Contents/Italy report|Italy report]]: Connecting the Arctic and Sicily through Wikimedia Projects
* [[outreach:GLAM/Newsletter/May 2026/Contents/Mexico report|Mexico report]]: How the cultural, governmental, educational, and museum sectors contributed in unison to Wikimedia projects
* [[outreach:GLAM/Newsletter/May 2026/Contents/Netherlands report|Netherlands report]]: Setting up your own platform (part 1): from wish to the creation of a wiki
* [[outreach:GLAM/Newsletter/May 2026/Contents/New Zealand report|New Zealand report]]: ESEAP Conference 2026 and the NZBSI WiR update
* [[outreach:GLAM/Newsletter/May 2026/Contents/Macedonia report|North Macedonia report]]: Wikimedia MKD Shares GLAM Experience at International Conferences 
* [[outreach:GLAM/Newsletter/May 2026/Contents/Poland report|Poland report]]: Partnerships, Participation, and Open Knowledge
* [[outreach:GLAM/Newsletter/May 2026/Contents/Serbia report|Serbia report]]: May in Wikimedia Serbia
* [[outreach:GLAM/Newsletter/May 2026/Contents/Spain report|Spain report]]: El Prado en femenino III
* [[outreach:GLAM/Newsletter/May 2026/Contents/Switzerland report|Switzerland report]]: Museum for Communication, Panel IMD, Wikipedia Day
* [[outreach:GLAM/Newsletter/May 2026/Contents/UK report|UK report]]: Early Qurans on the front page of Wikipedia
* [[outreach:GLAM/Newsletter/May 2026/Contents/Biodiversity Heritage Library report|Biodiversity Heritage Library report]]: Updates on work by BHLWiki Working Group members
* [[outreach:GLAM/Newsletter/May 2026/Contents/Special story|Special story]]: New tools for multilingual audio, video and metadata
* [[outreach:GLAM/Newsletter/May 2026/Contents/Memory of the World report|Memory of the World report]]: New articles in Urdu
* [[outreach:GLAM/Newsletter/May 2026/Contents/Events|Calendar]]: June's GLAM events

&lt;/div&gt;
|-
| style=&quot;font-family:Times New Roman; text-align:center; font-size:85%;&quot; | [[outreach:GLAM/Newsletter|Read this edition in full]] &amp;bull; [[outreach:GLAM/Newsletter/May 2026/Single|Single-page]]
|-
| valign=&quot;top&quot; colspan=&quot;2&quot; style=&quot;padding:0.5em; font-family:Times New Roman;text-align:center; font-size:85%;&quot; |
To assist with preparing the newsletter, please visit the [[outreach:GLAM/Newsletter/Newsroom|newsroom]]. Past editions may be viewed [[outreach:GLAM/Newsletter/Archives|here]].
|-
|}
|}
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[m:GLAM/Newsletter/About|About ''This Month in GLAM'']] · [[m:Global message delivery/Targets/GLAM|Subscribe/Unsubscribe]] · [[m:MassMessage|Global message delivery]] · [[:m:User:Romaine|Romaine]] 21:21, 12 June 2026 (UTC)&lt;/div&gt;
&lt;!-- Message sent by User:Romaine@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/GLAM&amp;oldid=30636889 --&gt;</text>
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  </page>
  <page>
    <title>Vehicle identification number</title>
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      <text bytes="803" sha1="5a082e40snss03zmwf1d2dcd2zutl7j" xml:space="preserve">[[File:Framenummer voorbeeld.jpg|thumb|Vehicle identification number]]

A '''vehicle identification number''' ('''VIN'''), also called a chassis number, is a unique code, including a serial number, used by the automotive industry to identify individual motor vehicles, towed vehicles, motorcycles, scooters and mopeds, as defined in ISO 3833.&lt;ref&gt;[[Wikipedia:Vehicle identification number]]&lt;/ref&gt;

== VIN Resources ==
* [https://www.edmunds.com/driving-tips/making-sense-of-your-vin.html Edmunds.com: Making sense of your VIN]
* [https://www.vindecoder.net/ VIN Decoder.net]
* [https://www.carfax.com/company/vehicle-identification-numbers-vins CARFAX VIN Decoder]
* [https://www.drivium.net Drivium VIN Decoder]
* [https://mothistoryfree.co.uk/ MOTHistoryFree VIN Decoder]

== References ==
{{Reflist}}</text>
      <sha1>5a082e40snss03zmwf1d2dcd2zutl7j</sha1>
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    <revision>
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      <text bytes="807" sha1="pfnejs5ric4bxaoi7mo4pox1j5gjias" xml:space="preserve">[[File:Framenummer voorbeeld.jpg|thumb|Vehicle identification number]]

A '''vehicle identification number''' ('''VIN'''), also called a chassis number, is a unique code, including a serial number, used by the automotive industry to identify individual motor vehicles, towed vehicles, motorcycles, scooters and mopeds, as defined in ISO 3833.&lt;ref&gt;[[Wikipedia:Vehicle identification number]]&lt;/ref&gt;

== VIN Resources ==
* [https://www.edmunds.com/driving-tips/making-sense-of-your-vin.html Edmunds.com: Making sense of your VIN]
* [https://www.vindecoder.net/ VIN Decoder.net]
* [https://www.carfax.com/company/vehicle-identification-numbers-vins CARFAX VIN Decoder]
* [https://www.drivium.net Drivium VIN Decoder]
* [https://www.mothistoryfree.co.uk/ MOTHistoryFree VIN Decoder]

== References ==
{{Reflist}}</text>
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  <page>
    <title>Category:Media reform to improve democracy</title>
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      <text bytes="9867" sha1="tdqt8j3qf35faiphx4dqfswruhlyprs" xml:space="preserve">A number of seemingly credible sources are describing an increase in political polarization worldwide. [[w:Maria Ressa|Maria Ressa]] describes how [[w:Rodrigo Duterte|Rodrigo Duterte]], former President of the [[w:Philippines|Philippines]] &quot;started ... with five hundred volunteers&lt;ref&gt;Ressa (2022, pp. 147-8).&lt;/ref&gt; (1) creating “sock puppets,” or fake accounts that attack or praise; (2) “mass reporting,” or organizing to negatively impact a targeted account; and (3) “astroturfing,” or fake posts or lies designed to look like grassroots support or interest.&lt;ref&gt;Ressa (2022, pp. 152-3).&lt;/ref&gt; These actions tricked the algorithms of social media companies like Facebook and Twitter into amplifying fraudulent messages including incitements to violence and criminal prosecutions based on trumped up charges. The results easily overwhelmed honest media. [[w:Leila de Lima|Leila de Lima]], a Senator and former Secretary of Justice of the Philippines, spent years  in pretrial detention before the charges were dropped for lack of evidence.&lt;ref&gt;Ressa (2022, p. 158ff) and Wikipedia, &quot;[[w:Leila de Lima|Leila de Lima]]&quot;, accessed 2024-07-22.&lt;/ref&gt; Ressa's news organization, [[w:Rappler|Rappler]].com, was ordered to close. Ressa herself was convicted on questionable charges. Both continued operating while the legal procedures against them were appealed.&lt;ref&gt;Ressa (2022, pp. 152-3) and Wikipedia, &quot;[[w:Maria Ressa|Maria Ressa]]&quot;, accessed 2024-07-22.&lt;/ref&gt; Ressa says similar procedures are making major contributions to the rise of fascism and far-right nationalist populists in the US, Europe, Turkey, India, Russia, and elsewhere.&lt;ref&gt;Ressa (2022, pp. 152-3).&lt;/ref&gt; [[w:H. R. McMaster|H. R. McMaster]], former President Trump's second National Security advisor, said that &quot;The internet and social media thus provided [Russia] with a low-cost, easy way to divide and weaken America from within.&quot;&lt;ref&gt;McMaster (2020, pp. 47-48).&lt;/ref&gt; The [[w:2021 Facebook leak|2021 Facebook leak]] documented how executives of [[w:Facebook|Facebook]] and [[w:Meta Platforms|Meta]] knowingly prioritized profits over action to limit incitements to violence, even facilitating the [[w:Rohingya genocide|Rohingya genocide]] in [[w:Myanmar|Myanmar]], because doing otherwise would have reduced their profits. 

This &quot;Category:Media reform to improve democracy&quot; include videos of experts and activists working this issue along with 29:00 mm:ss audio files submitted to a ''Media &amp; Democracy'' series syndicated on the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]&lt;ref&gt;&lt;!--Media &amp; Democracy on Audioport--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; plus text and space for moderated discussions. 

Some of this work is cited in the book on ''[[Media Literacy and You]]'', which is being written -- [[w:Crowdsourcing|crowdsourced]] -- to help humans better understand how they can counter the trend toward increasing political polarization and violence by talking politics, calmly, with respect and humility, with others with whom they may vehemently disagree, because the alternative is killing humans over misunderstanding. The goal is ''not'' to convince anyone that they are wrong. Rather it is to build relationships where humans can agree to disagree agreeably and collaborate to improve issues of common concern. 

== Table of episodes == 

{| class=&quot;wikitable sortable&quot;
|+ Episodes of &quot;Media &amp; Democracy&quot; for the [[w:List of Pacifica Radio stations and affiliates|Pacifica Radio Network]]
|-
! 
!! colspan=3 | Date !! 
|- 
! no. || recorded !! broadcasted on [[w:KKFI|KKFI]] !! released to Pacifica !! Episode 
|-
| 54 || 2026-06-18 || 2026-06-23 || 2026-06-27 || [[Wikimedia concerns with European copyright rules including AI and scientific research]]
|- 
| 53 || 2026-05-28 || 2026-06-09 || 2026-06-13 || [[Let's agree to disagree and seek common ground]]
|-
| 52 || 2026-05-14 || 2026-05-26 ||2026-05-30 || [[How women are centered and silenced in the major media]]
|-
| 51 || 2026-05-06 || 2026-05-12 || 2026-05-16 || [[Online platforms' effects on public health, safety and democracy]]
|-
| 50 || 2026-04-09	|| 2026-04-28 || 2026-05-02 || [[How US media threaten the health of all]]
|- 
| 49 || 2026-04-06 || 2026-04-14 || 2026-04-18 || [[News suppressed for those who control money for the media]] 
|-
| 48 || 2026-03-27 || 2026-03-31 || 2026-04-04 || [[Media and war]]
|-
| 47 || 2026-03-12 || 2026-03-17 || 2026-03-21 || [[Media literacy to dispel myths and improve public policy]]
|-
| 46 || 2026-02-26 || 2026-03-03 || 2026-03-07 || [[Concerns about media, especially in Germany]]
|-
| 45 || 2026-02-12 || 2026-02-17 || 2026-02-21 || [[Underserved serve themselves with low-power FM]]
|-
| 44 || 2026-01-30 || 2026-02-03 || 2026-02-07 || [[Conservative media are different]]
|-
| 43 || 2026-01-15 || 2026-01-20 || 2026-01-24 || [[Medill says you can help yourself by helping improve local media]] 
|-
| 42 || 2026-01-03 || 2026-01-06 || 2026-01-10 || [[Lisa Loving on media literacy and how you can report for your community]]
|- 
| 41 || 2015-12-11 || 2025-12-23 || 2025-12-27 || [[John Maxwell Hamilton on American propaganda]] 
|- 
| 40 || 2025-12-05 || 2025-12-09 || 2025-12-13 || [[You can better protect yourself from Big Tech]]
|- 
| 39 || 2025-11-20 || 2025-11-25 || 2025-11-29 || [[Differences between media outlets including coverage of Gaza]]
|- 
| 38 || 2025-11-06 || 2025-11-11 || 2025-11-15 || [[Media &amp; Democracy lessons for the future]]
|-
| 37 || 2025-10-23 || 2025-20-28 || 2025-11-01 || [[Media reform initiatives in West Africa]]  
|-
| 36 || 2025-10-03 || 2025-10-14 || 2025-10-18 || [[Seth Radwell says that the two Enlightenments tell us how to heal US political polarization]]
|-
| 35 || 2025-09-25 || 2025-09-30 || 2025-10-04 ||
[[Media Reform Coalition challenges anti-democratic media bias in the UK]]
|- 
| 34 || 2025-09-12 || 2025-09-16 || 2025-09-20 || [[Fighting back against the campaign of censorship and control]]
|- 
| 33 || 2025-08-28 || 2025-09-02 || 2025-08-06 || [[The role of the media in conflict]]
|- 
| 32 || 2025-07-31 || 2025-08-19 || 2025-08-21 || [[Evidence-informed public policy]]
|-
| 31 || 2025-08-01 || 2025-08-05 || 2025-08-09 || [[What the Left can learn from Fox]]
|-
| 30 || 2025-07-17 || 2025-07-22 || 2025-07-26 || [[Democratic delusions: Fix the media to fix democracy]]
|- 
| 29 || 2025-07-03 || 2025-07-08 || 2025-07-12 || [[News from Germany 1900-1945 and implications for today]]
|-
| 28 || 2025-06-12 || 2025-06-24 || 2025-06-28 || [[How news impacts democracy per USD Communications Professor Nik Usher]]
|-
| 27 || 2025-06-08 || 2025-06-10 || 2025-06-14 || [[Media concentration per Columbia History Professor Richard John]]
|- 
| 26 || 2025-05-21 || 2025-05-27 || 2025-05-31 || [[Dean Starkman and the watchdog that didn't bark]]
|- 
| 25 || 2025-05-08 || 2025-05-13 || 2025-05-17 || [[Freedom of the Press Foundation says...]]
|- 
| 24 || 2025-04-24 || 2025-04-29 || 2025-05-03 || [[Canadian journalist Marc Edge on media reform to improve democracy]]
|-
| 23 || 2025-04-10 || 2025-04-15 || 2025-04-19 || [[The value of indigenous and community radio]]
|- 
| 22 || 2025-03-28 || 2025-04-01 || 2025-04-05 || [[Trump ordered changes in public data]]
|-
| 21 || 2025-03-06 || 2025-03-11 || 2025-03-22 || [[Vulture capitalists destroying newspapers]]
|- 
| 20 || 2025-02-25 || 2025-02-25 || 2025-03-08 || [[Local newspapers limit malfeasance]]
|-
| 19 || 2025-02-06 || 2025-02-11 || 2025-02-22 || [[Palast says Trump lost, vote suppression won the 2024 elections]]
|-
| 18 || 2025-01-25 || 2025-02-04 || 2025-02-12 || [[Defend free speech hybrid town hall]]
|-
| 17 || 2025-01-13 || 2025-01-14 || 2025-01-25 || [[Media in the Syrian conflict]]
|-
| 16 || 2024-12-20 || 2024-12-31 || 2025-01-04 || [[HR 9495, the nonprofit-killer bill, per Michael Novick]]
|- 
| 15 || 2024-12-13 || 2024-12-24 || 2024-12-21 || [[Information is a public good per communications prof Pickard]]
|-
| 14 || 2024-12-02 || 2024-12-10 || 2024-12-07 || [[Media literacy for the Arab World per Ahmed Al-Rawi]]
|-
| 13 || 2024-11-21 || 2024-11-26 || 2024-11-23 || [[Thom Hartmann on The Hidden History of the American Dream]]
|-
| 12 || 2024-10-25 || 2024-11-05 || 2024-11-09 || [[Legal concerns of Wikimedia Europe]] 
|-
| 11 || 2024-10-26 || 2024-10-19 || 2024-10-27 || [[Project 2025 per Professor Brooks]]
|-
| 10 || 2024-10-01 || 2024-10-01 || 2024-10-12 || [[Jacob Ware on far-right terrorism in the US]]
|-
| 9 || 2024-09-13 || 2024-09-17 || 2024-09-29 || [[Dis- and misinformation and their threats to democracy]]
|- 
| 8 || 2024-09-11 || 2024-11-12 || 2024-09-14 || [[22nd Century Initiative]]
|- 
| 7 || 2024-08-22|| 2024-08-27 || 2024-08-31 || [[Global Project Against Hate &amp; Extremism (GPAHE)]]
|-
| 6 || 2024-08-19 || 2024-08-20 || 2024-08-24 || [[Facebook whistleblower Frances Haugen says]]
|-
| 5 || 2024-08-13 || 2024-08-13 || 2024-08-17 || [[Legal concerns of Free Press including Section 230]]
|-
| 4 || 2024-08-02 || 2024-08-06 || 2024-08-10 || [[How psychological and interpersonal processes are influenced by human-computer interactions]]
|-
| 3 || 2024-07-30 || 2024-07-30 || 2024-08-03 || [[Dean Baker on Internet companies threatening democracy internationally and how to fix that]]
|-
| 2 || 2021-04-29 || 2021-04-29 || 2021-05-16 || [[Media reform per Freepress.net]]
|- 
| 1 || 2021-02-23 || 2021-02-23 || 2021-03-17 ||[[Unrigging the media and the economy]]
|}

== Notes ==
{{reflist}}

== Bibliography == 

* &lt;!-- H. R. McMaster (2020) Battlegrounds: The Fight to Defend the Free World--&gt;{{cite Q|Q104774898}}
* &lt;!--Maria Ressa (2022) How to Stand Up To a Dictator--&gt;{{cite Q|Q117559286}}

[[Category:Interdisciplinary studies]]

[[Category:Political science]]
[[Category:Economics]]
[[Category:Freedom and abundance]]
[[Category:Videoconferences on media and democracy]]</text>
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[[Category:Class Lectures]]

I've asked ChatGPT about the Critical Theory on the Telegraph and it gave me the eight ways on how Critical Theory works with it. Before the telegraph was used by the highest power to communicate with other countries or businesses. Nobody could've had use of the telegraph until later on where the telephone was invented. Critical Theory even analyzed how it affected working conditions and employment patterns of telegraph operators. It also analyzed that the telegraph helped us understand a new language with Morse Code.https://chat.openai.com/share/8891bb96-ec2f-45fb-a82a-d6150c000622 Learn more about [https://morsecode.live/morse-code-chart/ Morse Code] symbols and patterns.
{{:Digital_Media_and_Information_in_Society/Theoretical_Frameworks}}
{{:Digital_Media_and_Information_in_Society/Theoretical_Frameworks_Early_Electronics_Telegraphic_Telephonic_Photographic_Phonographic}}
{{:Digital_Media_and_Information_in_Society/Theoretical_Frameworks_Summaries}}</text>
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    <title>Probability Dilation Theory</title>
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      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>Adding Convergence behavior section</comment>
      <origin>2815441</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38346" sha1="kgk9t11ug8w8jg5gam22yszhu1hekyg" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application showing that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Convergence behavior ==
Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, concentration effects, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.
=== Qualitative convergence classes ===
Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random weighting behavior.
In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.
Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.
Future investigation of PDT convergence behavior may include stability analysis, entropy-rate classification, stochastic convergence properties, fixed-point structure, and comparison with established dynamical systems and probabilistic evolution frameworks.
== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

whether a consistent discrete formulation of probability geometry can be constructed
whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles
whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions
== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application showing that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Convergence behavior ==
Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, concentration effects, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.
=== Qualitative convergence classes ===
Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random weighting behavior.
In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.
Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.
Future investigation of PDT convergence behavior may include stability analysis, entropy-rate classification, stochastic convergence properties, fixed-point structure, and comparison with established dynamical systems and probabilistic evolution frameworks.
== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

whether a consistent discrete formulation of probability geometry can be constructed
whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles
whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions
== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.


* Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656.

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application showing that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Convergence behavior ==
Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, concentration effects, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.
=== Qualitative convergence classes ===
Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random weighting behavior.
In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.
Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.
Future investigation of PDT convergence behavior may include stability analysis, entropy-rate classification, stochastic convergence properties, fixed-point structure, and comparison with established dynamical systems and probabilistic evolution frameworks.
== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

whether a consistent discrete formulation of probability geometry can be constructed
whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles
whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions
== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.


* Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656.

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application showing that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Convergence behavior ==
Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, concentration effects, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.
=== Qualitative convergence classes ===
Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random weighting behavior.
In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.
Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.
Future investigation of PDT convergence behavior may include stability analysis, entropy-rate classification, stochastic convergence properties, fixed-point structure, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

whether a consistent discrete formulation of probability geometry can be constructed
whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles
whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions
== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.


* Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656.

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application showing that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

whether a consistent discrete formulation of probability geometry can be constructed
whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles
whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions
== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.


* Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656.

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
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== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application showing that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

whether a consistent discrete formulation of probability geometry can be constructed
whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles
whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions
== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.


* Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656.

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential PDT transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

whether a consistent discrete formulation of probability geometry can be constructed
whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles
whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions
== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.


* Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656.

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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{{Original research}}
{{To be peer reviewed}}


== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential PDT transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.&lt;ref&gt;
Baryshnikov, Y., Cao, Y., Kahle, M., &amp; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19
&lt;/ref&gt;

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

whether a consistent discrete formulation of probability geometry can be constructed
whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles
whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions
== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.

* Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656.

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential PDT transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

whether a consistent discrete formulation of probability geometry can be constructed
whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles
whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions
== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

==Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:==
== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.

* Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656.

== Copyright and licensing ==

Text and original figures © Howard Richardson.

Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Reuse permitted with attribution.</text>
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    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
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      <comment>/* The 24-cell */</comment>
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      <text bytes="74960" sha1="6zu4157r6jzpt8m54unme3mw3u4j1t9" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
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'' taking its edge planes to each other. 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 another 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{\pi}{2}/2)=\sqrt{2}&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. We can rotate the 600-cell isoclinically in two completely orthogonal invariant square planes containing 16-cell edges, 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 does not visit the vertex positions of the other 16-cells. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; chords form a circular helix of twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct rotation in completely orthogonal great square invariant planes, over the &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; chord. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. In each 90° isoclinic rotational displacement it takes every great square plane to a great square plane in another 16-cell. The rotational curve over each 90° &lt;math&gt;r_7&lt;/math&gt; chord makes seven 12° turns. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;14\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.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in hexagon 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 hexagon rotation characteristic of the 24-cell must be 5{24/10}=10{12/5} over &lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt; chords. Four {30/9}=3{10/3} over &lt;math&gt;r_9=\phi&lt;/math&gt; chords in the illustration is a distinct rotation arising in the 600-cell, one we shouldn't be illustrating here, unless we're going to illustrate all the non-edge 24-cell and 600-cell rotations.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;math&gt;r_9=\phi&lt;/math&gt;]]
We can also rotate the 600-cell isoclinically in decagon 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'' taking its edge planes to each other. 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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75245" sha1="6cpuye79dh131jmp7dlozj3f7qrtbop" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
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'' taking its edge planes to each other. 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 another 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{\pi}{2}/2)=\sqrt{2}&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_10=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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 does not visit vertex positions of other 16-cells. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement it takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt; isocline chords, 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 vertex positions of other 24-cells. The &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} &lt;math&gt;r_9&lt;/math&gt; edges.]][Four {30/9}=3{10/3} over &lt;math&gt;r_9=\phi&lt;/math&gt; chords in the illustration is a distinct rotation arising in the 600-cell, one we shouldn't be illustrating here, unless we're going to illustrate all the non-edge 24-cell and 600-cell rotations.]

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;math&gt;r_9=\phi&lt;/math&gt;]]
We can also rotate the 600-cell isoclinically in decagon 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'' taking its edge planes to each other. 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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75279" sha1="kem1wozkvfa0rn4jv47o4c4v73fr58u" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and 3 Clifford parallel invariant hexagon planes. 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'' taking its edge planes to each other. 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 another 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{\pi}{2}/2)=\sqrt{2}&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_10=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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 does not visit vertex positions of other 16-cells. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement it takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt; isocline chords, 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 vertex positions of other 24-cells. The &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} &lt;small&gt;&lt;math&gt;r_9=\phi&lt;/math&gt;&lt;/small&gt;]][Four {30/9}=3{10/3} over &lt;math&gt;r_9=\phi&lt;/math&gt; chords in the illustration is a distinct rotation arising in the 600-cell, one we shouldn't be illustrating here, unless we're going to illustrate all the non-edge 24-cell and 600-cell rotations.]

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_8=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in an invariant decagon central plane and 11 Clifford parallel invariant decagon planes, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. Successive &lt;math&gt;\sqrt{1}&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4}  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;  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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75281" sha1="1qly3s4vyck3ghi8d5hmsi6zjzbad8v" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and 3 Clifford parallel invariant hexagon planes. 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'' taking its edge planes to each other. 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 another 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{\pi}{2}/2)=\sqrt{2}&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_10=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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 does not visit vertex positions of other 16-cells. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement it takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt; isocline chords, 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 vertex positions of other 24-cells. The &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} &lt;small&gt;&lt;math&gt;r_9=\phi&lt;/math&gt;&lt;/small&gt;]][Four {30/9}=3{10/3} over &lt;math&gt;r_9=\phi&lt;/math&gt; chords in the illustration is a distinct rotation arising in the 600-cell, one we shouldn't be illustrating here, unless we're going to illustrate all the non-edge 24-cell and 600-cell rotations.]

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_8=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in an invariant decagon central plane and 11 Clifford parallel invariant decagon planes, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. Successive &lt;math&gt;\sqrt{1}&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4}  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;  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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75483" sha1="t5nt82kfmxibx6nqfr0fmvk42j9bywc" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. 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'' taking its edge planes to each other. 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 another 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}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&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{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&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}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_10=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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 does not visit vertex positions of other 16-cells. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement it takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt; isocline chords, 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 vertex positions of other 24-cells. The &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} &lt;small&gt;&lt;math&gt;r_9=\phi&lt;/math&gt;&lt;/small&gt;]][Four {30/9}=3{10/3} over &lt;math&gt;r_9=\phi&lt;/math&gt; chords in the illustration is a distinct rotation arising in the 600-cell, one we shouldn't be illustrating here, unless we're going to illustrate all the non-edge 24-cell and 600-cell rotations.]

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_8=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. Successive &lt;math&gt;\sqrt{1}&lt;/math&gt; chords are edges of different 24-cells. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4}  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;  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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75583" sha1="a0sadibk2dcoz5m3y86qaoq7te3gp84" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. 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'' taking its edge planes to each other. 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 another 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}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&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{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&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}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_10=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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 does not visit vertex positions of other 16-cells. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement it takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, 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 vertex positions of other 24-cells. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} &lt;small&gt;&lt;math&gt;r_9=\phi&lt;/math&gt;&lt;/small&gt;]][Four {30/9}=3{10/3} over &lt;math&gt;r_9=\phi&lt;/math&gt; chords in the illustration is a distinct rotation arising in the 600-cell, one we shouldn't be illustrating here, unless we're going to illustrate all the non-edge 24-cell and 600-cell rotations.]

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_8=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4} that visits each 600-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 600-cell'' taking its edge planes to each other. 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. 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;r_5&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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75228" sha1="170yae219019rgtqx0lcpke0alp8asa" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. 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'' taking its edge planes to each other. 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 another 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}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&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{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&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}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_10=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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 does not visit vertex positions of other 16-cells. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement it takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, 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 vertex positions of other 24-cells. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_8=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4} that visits each 600-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 600-cell'' taking its edge planes to each other. 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. 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;r_5&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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75200" sha1="sfh0s203r0j56y5a69towmqrnuo7dk3" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. 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'' taking its edge planes to each other. 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 another 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}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&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{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&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}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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, and it does not visit other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, 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, and it does not visit other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4} that visits each 600-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 600-cell'' taking its edge planes to each other. 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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. 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;r_5&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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75117" sha1="q79289aa2uv56n0qcntdue77dwqhufr" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. 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'' taking its edge planes to each other. 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 another 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}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&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{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&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}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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, and it does not visit other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, 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, and it does not visit other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other, also 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;. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. 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;r_5&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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75126" sha1="pyspibuohtpixxjsrj31dg1sbv2ngg9" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. 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'' taking its edge planes to each other. 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 another 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}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&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{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&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}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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, and it does not visit other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, 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, and it does not visit other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other, also 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;. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. 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;r_5&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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75160" sha1="k10v5kmvhhczost535s7wf33vy752wg" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. 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'' taking its edge planes to each other. 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 another 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}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&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{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&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}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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, and does not visit other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, 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, and does not visit other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other, also 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;. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. 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 of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&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 complete isoclinic revolution 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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. 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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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>2815460</id>
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      <text bytes="75215" sha1="i7zerzfc84ufw7ltw8v2ndnki4zeuvu" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. 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'' taking its edge planes to each other. 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 another 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}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&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{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&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}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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, and does not visit other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, 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, and does not visit other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its edges, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other, also 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;. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. 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 of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&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 complete isoclinic revolution 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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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="75122" sha1="988yl76y8eg9ci3jpdpf1mdx06kofaw" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also 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;. 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 hexagonal 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. 

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 another 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}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&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{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&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}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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, and does not visit other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, 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, and does not visit other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. 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 of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&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 complete isoclinic revolution 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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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>2815470</id>
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      <text bytes="75118" sha1="tj48m4eh7e5aiqy0c88qy8rao4x3owg" 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 and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; 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|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} 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 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w: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 tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== 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 rotation of the 16-cell taking its edge planes to each other, 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} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
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} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant hexagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' taking its edge planes to each other, also 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;. 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 hexagonal 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. 

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 another 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}=\text{24-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&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{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_1&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&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}=\text{24-cell-}r_4&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&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_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in completely orthogonal invariant square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, 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, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that visits each 600-cell vertex once.

[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} &lt;small&gt;&lt;math&gt;r_7=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation in completely orthogonal great square invariant planes, over &lt;math&gt;r_7=\sqrt{2}&lt;/math&gt; isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The &lt;math&gt;r_7&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_1&lt;/math&gt; chord.  The rotational curve over each 90° &lt;math&gt;r_1&lt;/math&gt; chord makes just one 45° turn. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;\tfrac{15\pi}{2}&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.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically in invariant hexagon planes over &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, 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, without visiting other vertex positions. The &lt;math&gt;r_{11}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each 60° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that visits each 600-cell vertex once. 

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_5=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{1}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' taking its edge planes to each other. Its Clifford polygon is a skew {15/4} pentadecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The &lt;math&gt;r_5&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord. Successive &lt;math&gt;r_5&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_5&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 parallel strands 4{30/8}=8{15/4} that visits each 600-cell vertex once.    

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 in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. 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 of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&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 complete isoclinic revolution 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 has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-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. 

...

== 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 isocclinic 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 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons 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>User:Dc.samizdat/Golden chords of the 120-cell/sandbox</title>
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        <username>Dc.samizdat</username>
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      <text bytes="66671" sha1="d1w7r394uzatvbv5n4rf2dm1xsgg45r" xml:space="preserve">== The 600 cell ==
[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} &lt;small&gt;&lt;math&gt;r_9=\phi&lt;/math&gt;&lt;/small&gt;]][Four {30/9}=3{10/3} over &lt;math&gt;r_9=\phi&lt;/math&gt; chords in the illustration is a distinct rotation arising in the 600-cell, one we shouldn't be illustrating here, unless we're going to illustrate all the non-edge 24-cell and 600-cell rotations.]


== Radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 120-cell ==
{| 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|roots
!colspan=7|Chord lengths of the &lt;math&gt;\sqrt{2}&lt;/math&gt; 120-cell
|-
!colspan=5|length &lt;math&gt;c_t&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;\sqrt{2}&lt;/math&gt;
!colspan=2|length &lt;math&gt;c_t \times \phi^2/\sqrt{2}&lt;/math&gt;&lt;br&gt;in 120-cell of edge &lt;math&gt;1/\sqrt{2}&lt;/math&gt;, radius &lt;math&gt;c_8=\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,2}&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;\{30\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,2}-c_{2,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(3-\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.381966&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.145898}&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_{2,2}&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;\{\frac{30}{2}\}&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 \{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,2}-c_{4,2}\right)&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;\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_{3,2}&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;\{\frac{30}{3}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \{\frac{10}{3}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5}-1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.874032&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{2}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.763932}&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_{4,2}&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;\{\frac{60}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,2}}{\sqrt{2}}&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;\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_{5,2}&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;\{\frac{30}{4}\}&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 \{\frac{15}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{3} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.07047&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.1459}&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_{6,2}&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;\{\frac{120}{17}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,2}&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;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{\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[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_{7,2}&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;\{\frac{20}{3}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\frac{4}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.32813&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.76393}&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_{8,2}&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;\{\frac{30}{5}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{6\}&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;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;\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_{9,2}&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;\{\frac{40}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.54336&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.38197}&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_{10,2}&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;\{\frac{60}{11}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,2}&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{\phi ^2}&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;\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_{11,2}&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;\{\frac{30}{6}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{2 \sqrt[4]{5}}{\sqrt{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.66251&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 (3-\phi )}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.76393}&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_{12,2}&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;\{\frac{24}{5}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,2}&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{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_{13,2}&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;\{\frac{60}{13}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.83901&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.38197}&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_{14,2}&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;\{\frac{40}{9}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,2}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{1+\sqrt{5}}}{\sqrt{2}}&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[4]{5} \sqrt{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5} \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;\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_{15,2}&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;\{\frac{30}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,2}&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;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 \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_{16,2}&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;\{\frac{120}{29}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.09331&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.38197}&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_{17,2}&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;\{\frac{120}{31}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.14896&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.61803}&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_{18,2}&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;\{\frac{30}{8}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{15}{4}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.23607&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{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_{19,2}&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;\{\frac{30}{9}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{10}{3}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,2}+c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{\sqrt{2}}&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;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 (1+\phi )}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.23607}&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_{20,2}&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;\{\frac{120}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.31991&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.38197}&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_{21,2}&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;\{\frac{60}{19}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.37024&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 \left(\frac{5}{2}+\frac{1}{1+\sqrt{5}}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 \left(\frac{5}{2}+\frac{1}{1+\sqrt{5}}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.61803}&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_{22,2}&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;\{\frac{30}{10}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.44949&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.}&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_{23,2}&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;\{\frac{120}{41}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5+\frac{4}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.49721&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 \left(4-\frac{\psi }{2 \phi }\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.23607}&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_{24,2}&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;\{\frac{20}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.57255&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.61803}&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_{25,2}&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;\{\frac{30}{11}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{30}{11}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(7+3 \sqrt{5}\right)}&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;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.8541}&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_{26,2}&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;\{\frac{12}{5}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.64575&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.}&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_{27,2}&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;\{\frac{30}{12}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.68999&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 (2+\phi )}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.23607}&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_{28,2}&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;\{\frac{30}{13}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{30}{13}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.76008&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.61803}&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_{29,2}&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;\{\frac{30}{14}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{15}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{3} \left(1+\sqrt{5}\right)&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;\sqrt{3} \phi &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;\sqrt{7.8541}&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_{30,2}&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;\{\frac{30}{15}\}&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 c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.82843&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.}&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;
|}

The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.

== Radius &lt;math&gt;\phi&lt;/math&gt; 120-cell ==
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=9|Chord lengths of the &lt;math&gt;\phi&lt;/math&gt; 120-cell
|-
!&lt;math&gt;c_t&lt;/math&gt;
!arc
!&lt;math&gt;\frac{k}{d}&lt;/math&gt;
!colspan=4|length &lt;math&gt;c_t&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;\phi&lt;/math&gt;
!colspan=2|length &lt;math&gt;c_t\sqrt{2}&lt;/math&gt;&lt;br&gt;in 120-cell of edge &lt;math&gt;1/\phi&lt;/math&gt;, radius &lt;math&gt;c_8=\sqrt{2}\phi&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_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;30&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;\sqrt{\frac{2}{\left(1+\sqrt{5}\right)^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;\frac{1}{\phi }&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;c_2&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;15&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;\sqrt{\frac{1}{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;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_3&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;10&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;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{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;c_4&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;\frac{60}{7}&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;\sqrt{\frac{1}{8} \left(1+\sqrt{5}\right)^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;\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_5&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;\frac{15}{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{\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{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;c_6&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;\frac{120}{17}&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{\frac{1}{4} \sqrt{5} \left(1+\sqrt{5}\right)}&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{2} \sqrt{\frac{1}{\phi }}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi ^3}}{\phi }&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;c_7&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;\frac{20}{3}&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{\frac{1}{8} \left(1+\sqrt{5}\right) \left(-1+3 \sqrt{5}\right)}&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{\phi  \sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\psi  \phi ^3}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.14896&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_8&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;6&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{\frac{1}{4} \left(1+\sqrt{5}\right)^2}&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{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_9&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;\frac{40}{7}&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{\frac{1}{8} \left(1+\sqrt{5}\right) \left(1+3 \sqrt{5}\right)}&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;\frac{\phi  \sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^3}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.49721&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{10}&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;\frac{60}{11}&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;\sqrt{\frac{1}{32} \left(1+\sqrt{5}\right)^4}&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;\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_{11}&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;5&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;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(3+\frac{1}{2} \left(-1-\sqrt{5}\right)\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{3-\phi } \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.68999&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{12}&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;\frac{24}{5}&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{\frac{3}{8} \left(1+\sqrt{5}\right)^2}&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{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_{13}&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;\frac{60}{13}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{16} \left(9-\sqrt{5}\right) \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.10406&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.13657&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{14}&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;\frac{40}{9}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.73607}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{16} \sqrt{5} \left(1+\sqrt{5}\right)^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;\sqrt[4]{5} \phi ^{3/2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \phi  \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;8.05748&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{15}&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;\frac{30}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.23607}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)^2}&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;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.23607&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{16}&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;\frac{120}{29}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.73607}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{16} \left(11-\sqrt{5}\right) \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.39501&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.38705&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{17}&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;\frac{120}{31}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.04508}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4+\frac{1}{4} \left(-9+\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.45868&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4+\frac{1}{4} \left(\sqrt{5}-9\right)} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\psi  \phi ^5}}{\phi }&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_{18}&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;\frac{15}{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.54508}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{8} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.55834&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5} \sqrt{\phi ^4}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.61803&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{19}&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;\frac{10}{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.8541}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{16} \left(1+\sqrt{5}\right)^4}&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;\phi ^2&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_{20}&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;\frac{120}{37}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.04508}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4-\frac{1}{8} \left(1+\sqrt{5}\right)^2\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.65426&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{4-\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.75369&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{21}&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;\frac{60}{19}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.3541}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4-\frac{1+3 \sqrt{5}}{2 \left(1+\sqrt{5}\right)}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.71184&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{4-\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{8-\frac{\phi }{\chi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.45479&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{22}&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;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.8541}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{4} \left(1+\sqrt{5}\right)^2}&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;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.96336&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{23}&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;\frac{120}{41}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.16312}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4-\frac{-1+3 \sqrt{5}}{2 \left(1+\sqrt{5}\right)}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.85712&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^5}}{\phi }&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_{24}&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;\frac{20}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.66312}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4-\frac{\sqrt{5}}{1+\sqrt{5}}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.94332&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\sqrt{5}}{2 \phi }} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.16248&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{25}&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;\frac{30}{11}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.97214}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{128} \left(1+\sqrt{5}\right)^6}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.99535&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^3}{\sqrt{2}}&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_{26}&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;\frac{12}{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.16312}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{8} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.02706&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.28092&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{27}&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;\frac{5}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.47214}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(2+\frac{1}{2} \left(1+\sqrt{5}\right)\right)}&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;\phi  \sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{2 \phi +4}&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_{28}&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;\frac{30}{13}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.97214}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4-\frac{2}{\left(1+\sqrt{5}\right)^2}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.15787&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{1}{2 \phi ^2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.4659&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{29}&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;\frac{15}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.2812}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{32} \left(1+\sqrt{5}\right)^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.20642&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{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_{30}&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;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.4721}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.23607&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.57649&lt;/math&gt;&lt;/small&gt;
|}

== Radius &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; 120-cell ==
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=9|Chord lengths of the &lt;math&gt;\sqrt{3}&lt;/math&gt; 120-cell
|-
!&lt;math&gt;c_t&lt;/math&gt;
!arc
!&lt;math&gt;\frac{k}{d}&lt;/math&gt;
!colspan=4|length &lt;math&gt;c_t&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;\sqrt{3}&lt;/math&gt;
!colspan=2|length &lt;math&gt;c_t \times c_8/\sqrt{3}&lt;/math&gt;&lt;br&gt;in 120-cell of edge &lt;math&gt;1/\sqrt{3}&lt;/math&gt;, radius &lt;math&gt;c_8=\sqrt{\frac{2}{3}}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_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;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.218847}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{24}{\left(1+\sqrt{5}\right)^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.467811&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.57735&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_2&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;15&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{\frac{6}{\left(1+\sqrt{5}\right)^2}}&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;\frac{\phi }{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.934172&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_3&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;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.1459}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{12}{\left(1+\sqrt{5}\right)^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.07047&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{3}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.32112&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_4&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;\frac{60}{7}&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{\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;\frac{\phi ^2}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51152&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_5&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;\frac{15}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.71885}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{18}{\left(1+\sqrt{5}\right)^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.31105&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{3}{\sqrt{2} \phi }&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_6&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;\frac{120}{17}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.07295}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \sqrt{5}}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.43977&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \sqrt[4]{5} \sqrt{\frac{1}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi ^3}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7769&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_7&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;\frac{20}{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.6459}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \left(-1+3 \sqrt{5}\right)}{2 \left(1+\sqrt{5}\right)}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.62662&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \sqrt{\frac{\psi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\psi  \phi ^3}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.0075&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_8&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;6&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;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{\frac{2}{3}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.13762&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_9&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;\frac{40}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.57295}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \left(1+3 \sqrt{5}\right)}{2 \left(1+\sqrt{5}\right)}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.89022&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \sqrt{\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^3}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.33283&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{10}&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;\frac{60}{11}&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{\frac{3}{8} \left(1+\sqrt{5}\right)^2}&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;\frac{\phi ^3}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.44569&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{11}&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;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.1459}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3 \left(3+\frac{1}{2} \left(-1-\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.03615&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{3}} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.51292&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{12}&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;\frac{24}{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{9}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.12132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{3}{\sqrt{2}}&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_{13}&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;\frac{60}{13}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.07295}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.25232&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{3 \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{6} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.06175&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{14}&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;\frac{40}{9}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{4} \sqrt{5} \left(1+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.29456&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \sqrt[4]{5} \sqrt{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \phi ^2 \sqrt{\phi ^5}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.52708&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{15}&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;\frac{30}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.44949&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{2 \phi ^2}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.02305&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{16}&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;\frac{120}{29}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.57295}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.56378&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{3 \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{6} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.16409&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{17}&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;\frac{120}{31}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3 \left(4+\frac{1}{4} \left(-9+\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.63193&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3 \left(4+\frac{1}{4} \left(\sqrt{5}-9\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\psi  \phi ^5}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.2482&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{18}&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;\frac{15}{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{15}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.73861&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{15}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{3}} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.37987&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{19}&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;\frac{10}{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.8541}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{4} \left(1+\sqrt{5}\right)^2}&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;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{3}} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.45874&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{20}&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;\frac{120}{37}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.07295}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3 \left(4-\frac{1}{8} \left(1+\sqrt{5}\right)^2\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.84129&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \sqrt{4-\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\phi ^2}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.50659&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{21}&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;\frac{60}{19}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3 \left(4-\frac{1+3 \sqrt{5}}{2 \left(1+\sqrt{5}\right)}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.90294&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \sqrt{4-\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\frac{\phi }{\chi }}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.16154&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{22}&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;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3&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_{23}&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;\frac{120}{41}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.3541}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3 \left(4-\frac{-1+3 \sqrt{5}}{2 \left(1+\sqrt{5}\right)}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.05845&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^5}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.77459&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{24}&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;\frac{20}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3 \left(4-\frac{\sqrt{5}}{1+\sqrt{5}}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.15072&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \sqrt{4-\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.88847&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{25}&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;\frac{30}{11}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.2812}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{32} \left(1+\sqrt{5}\right)^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.20642&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;\frac{\phi ^4}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.95722&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{26}&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;\frac{12}{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{21}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.24037&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{21}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{3}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.99911&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{27}&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;\frac{5}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.8541}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3 \left(2+\frac{1}{2} \left(1+\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.29456&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{2 \phi +4}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.06599&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{28}&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;\frac{30}{13}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{11.4271}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3 \left(4-\frac{2}{\left(1+\sqrt{5}\right)^2}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.38039&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \sqrt{4-\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.17192&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{29}&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;\frac{15}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{11.7812}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{9}{8} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.43237&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{3 \phi }{\sqrt{2}}&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_{30}&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;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{12.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{12}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.4641&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{\frac{2}{3}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.27523&lt;/math&gt;&lt;/small&gt;
|}

== Radius &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt; 120-cell ==
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=9|Chord lengths of the &lt;math&gt;\sqrt{5}&lt;/math&gt; 120-cell
|-
!&lt;math&gt;c_t&lt;/math&gt;
!arc
!&lt;math&gt;\frac{k}{d}&lt;/math&gt;
!colspan=4|length &lt;math&gt;c_t&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;\sqrt{5}&lt;/math&gt;
!colspan=2|length &lt;math&gt;c_t \times c_8/\sqrt{5}&lt;/math&gt;&lt;br&gt;in 120-cell of edge &lt;math&gt;1/\sqrt{5}&lt;/math&gt;, radius &lt;math&gt;c_8 = \sqrt{\frac{2}{5}} \phi ^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_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;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.364745}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{40}{\left(1+\sqrt{5}\right)^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.603941&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{5}{2}}}{\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.447214&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_2&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;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.954915}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{10}{\left(1+\sqrt{5}\right)^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.977198&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{5}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.723607&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_3&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;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.90983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{20}{\left(1+\sqrt{5}\right)^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.38197&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{5}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.02333&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_4&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;\frac{60}{7}&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{\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;\frac{\phi ^2}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17082&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_5&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;\frac{15}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.86475}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{30}{\left(1+\sqrt{5}\right)^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.69256&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{15}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{5}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.25332&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_6&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;\frac{120}{17}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.45492}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5 \sqrt{5}}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85874&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{5^{3/4} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\phi ^3}}{\sqrt[4]{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.37638&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_7&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;\frac{20}{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.40983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5 \left(-1+3 \sqrt{5}\right)}{2 \left(1+\sqrt{5}\right)}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.09996&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} \sqrt{\frac{\psi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\psi  \phi ^3}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.555&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_8&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;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.23607&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{5}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.65579&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_9&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;\frac{40}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.95492}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5 \left(1+3 \sqrt{5}\right)}{2 \left(1+\sqrt{5}\right)}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.44027&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} \sqrt{\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^3}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.807&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{10}&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;\frac{60}{11}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.54508}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{8} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.55834&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^3}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.89443&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{11}&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;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.90983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(3+\frac{1}{2} \left(-1-\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.62866&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{5}} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.9465&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{12}&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;\frac{24}{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{15}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.73861&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{15}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{5}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.02792&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{13}&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;\frac{60}{13}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.45492}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.90773&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5 \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{10} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.822431&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{14}&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;\frac{40}{9}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.04508}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{4} \sqrt{5} \left(1+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.25325&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5^{3/4} \sqrt{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{\phi ^5}}{\sqrt[4]{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.83045&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{15}&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;\frac{30}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.16228&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{2 \phi ^2}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.34164&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{16}&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;\frac{120}{29}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.9549}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.30982&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5 \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{10} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.4509&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{17}&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;\frac{120}{31}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{11.5451}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4+\frac{1}{4} \left(-9+\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.39781&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4+\frac{1}{4} \left(\sqrt{5}-9\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\psi  \phi ^5}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.51605&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{18}&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;\frac{15}{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{12.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{25}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.53553&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{5}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi ^4}&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_{19}&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;\frac{10}{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{13.0902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{4} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{5}} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.67912&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{20}&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;\frac{120}{37}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{13.4549}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4-\frac{1}{8} \left(1+\sqrt{5}\right)^2\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.66809&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{4-\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\phi ^2}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.71619&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{21}&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;\frac{60}{19}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{14.0451}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4-\frac{1+3 \sqrt{5}}{2 \left(1+\sqrt{5}\right)}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.74768&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{4-\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\frac{\phi }{\chi }}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.22352&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{22}&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;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{15.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{15}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.87298&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{15}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{6}{5}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.86791&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{23}&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;\frac{120}{41}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{15.5902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4-\frac{-1+3 \sqrt{5}}{2 \left(1+\sqrt{5}\right)}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.94844&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^5}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.92379&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{24}&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;\frac{20}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{16.5451}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4-\frac{\sqrt{5}}{1+\sqrt{5}}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.06756&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{4-\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.012&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{25}&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;\frac{30}{11}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{17.1353}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{32} \left(1+\sqrt{5}\right)^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.13948&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^4}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.06525&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{26}&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;\frac{12}{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{17.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{35}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.1833&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{35}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{5}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.0977&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{27}&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;\frac{5}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{18.0902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(2+\frac{1}{2} \left(1+\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.25325&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{2 \phi +4}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.1495&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{28}&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;\frac{30}{13}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{19.0451}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4-\frac{2}{\left(1+\sqrt{5}\right)^2}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.36407&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{4-\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.23156&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{29}&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;\frac{15}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{19.6353}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{15}{8} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.43117&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{15}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{5}} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.28124&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{30}&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;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{20.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{20}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.47214&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{\frac{2}{5}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.31158&lt;/math&gt;&lt;/small&gt;
|}

== Steinbach's golden chords ==
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=11|
|-
!
!
!colspan=2 align=left|pentagon {5}
!colspan=2 align=left|heptagon {7}
!colspan=2 align=left|nonagon {9}
!colspan=2 align=left|hendecagon {11}
!
|-
|
|
|colspan=2 align=left|
&lt;small&gt;&lt;math&gt;\phi=\frac{1}{2}(1 + \sqrt{5}) \approx 1.618034&lt;/math&gt;&lt;/small&gt;
|colspan=2 align=left|
&lt;small&gt;&lt;math&gt;\rho=2\cos{\pi/7} \approx 1.80194&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\sigma=\rho^2 - 1 \approx 2.24698&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
|colspan=2 align=left|
&lt;small&gt;&lt;math&gt;\alpha=2\cos{\pi/9} \approx 1.87939&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\beta=&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\gamma=&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
|colspan=2 align=left|
&lt;small&gt;&lt;math&gt;\theta=2\cos{\pi/11} \approx 1.91899&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\kappa=&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\lambda=&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\mu=&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
|
|}

{{Efn|&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi=\frac{1}{2}(\sqrt{5} + 1) \approx 1.618034&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\Phi=\frac{1}{2}(\sqrt{5} - 1) \approx 0.618034&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\chi=\frac{1}{2}(3\sqrt{5} + 1) \approx 3.854102&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\psi=\frac{1}{2}(3\sqrt{5} - 1) \approx 2.854102&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
|name=phi constants}}

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=20|Chord lengths of the 120-cell
|-
|1
|1.
|-                        
|2
|1.
|-                        
|3
|1.
|-                       
|4
|1.
|2
|90°
|1.00
|-                     
|5
|1.
|&lt;math&gt;\phi&lt;/math&gt;
|144°
|1.62
|-                     
|6
|1.
|3
|120°
|1.73
|4
|180°
|2.00 
|-                 
|7
|1.
|&lt;math&gt;\rho&lt;/math&gt;
|102.9°
|1.80
|&lt;math&gt;\sigma&lt;/math&gt;
|154.3°
|2.25
|}
{{Void|                
8	1.	4₂	90 °	1.85	4₃	135 °	2.41                  
9	1.	&lt;math&gt;alpha&lt;/math&gt;	80 °	1.88	&lt;math&gt;beta&lt;/math&gt;	120 °	2.53	&lt;math&gt;gamma&lt;/math&gt;	160 °	2.88               
10	1.	5₂	72 °	1.90	5₃	108 °	2.62	5₄	144 °	3.08               
11	1.	&lt;math&gt;theta&lt;/math&gt;	65.5 °	1.92	&lt;math&gt;kappa&lt;/math&gt;	98.2 °	2.68	&lt;math&gt;lambda&lt;/math&gt;	130.9 °	3.23	&lt;math&gt;mu&lt;/math&gt;	163.6 °	3.51            
12	1.	6₂	60 °	1.93	6₃	90 °	2.73	6₄	120 °	3.35	6₅	150 °	3.73            
13	1.	_	55.4 °	1.94	_	83.1 °	2.77	_	110.8 °	3.44	_	138.5 °	3.91	_	166.2 °	4.15	_	166.2 °	4.15      
14	1.	_₂	51.4 °	1.95	_₃	77.1 °	2.80	_₄	102.9 °	3.51	_₅	128.6 °	4.05	_₆	154.3 °	4.38	_₇	180 °	4.49      
15	1.	_	48 °	1.96	_	72 °	2.83	_	96 °	3.57	_	120 °	4.17	_	144 °	4.57	_	168 °	4.78	_	168 °	4.78	_	144 °	4.57
16	1.	_₂	45 °	1.96	_₃	67.5 °	2.85	_₄	90 °	3.62	_₅	112.5 °	4.26	_₆	135 °	4.74	_₇	157.5 °	5.03	_₈	180 °	5.13	_₉	157.5 °	5.03
}}
{{Void|
chordL :={
{{&quot;1₂&quot;}},
{{&quot;1₁&quot;}},
{{&quot;3₂&quot;}},
{{&quot;2₁&quot;}},
{{&quot;5₂&quot;},{ &quot;5₄&quot;}},
{{&quot;3₁&quot;},{ &quot;3₂&quot;}},
{{&quot;7₂&quot;},{&quot;7₄&quot;},{&quot;7₆&quot;}},
{{&quot;4₁&quot;},{ &quot;4₂&quot;},{ &quot;4₃&quot;}},
{{&quot;9₂&quot;}, {&quot;9₄&quot;},{&quot;9₆&quot;},{&quot;9₈&quot;}},
{{&quot;5₁&quot;}, { &quot;5₂&quot;},{ &quot;5₃&quot;},{ &quot;5₄&quot;}},
{{&quot;11₂&quot;},{&quot;11₄&quot;},{&quot;11₆&quot;},{&quot;11₈&quot;},{&quot;11₁₀&quot;}} ,
{{&quot;6₁&quot;},{ &quot;6₂&quot;},{ &quot;6₃&quot;},{ &quot;6₄&quot;},{ &quot;6₅&quot;}},
{{&quot;13₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}, { &quot;_&quot;}},
{{&quot;7₁&quot;},{ &quot;7₂&quot;},{ &quot;7₃&quot;},{ &quot;7₄&quot;},{ &quot;7₅&quot;}, { &quot;7₆&quot;}},
{{&quot;15₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}, { &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;8₁&quot;},{ &quot;8₂&quot;},{ &quot;8₃&quot;},{ &quot;8₄&quot;},{ &quot;8₅&quot;},{ &quot;8₆&quot;},{ &quot;8₇&quot;}},
{{&quot;17₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;9₁&quot;},{ &quot;9₂&quot;},{ &quot;9₃&quot;},{ &quot;9₄&quot;},{ &quot;9₅&quot;},{ &quot;9₆&quot;},{ &quot;9₇&quot;},{ &quot;9₈&quot;}},
{{&quot;19₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;10₁&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;21₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;11₁&quot;},{ &quot;11₂&quot;},{ &quot;11₃&quot;},{ &quot;11₄&quot;},{ &quot;11₅&quot;},{ &quot;11₆&quot;},{ &quot;11₇&quot;},{ &quot;11₈&quot;},{ &quot;11₉&quot;},{ &quot;11₁₀&quot;}},
{{&quot;23₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;12₁&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;25₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;13₁&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;27₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;15₁&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;29₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;16₁&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}}
}}

== Golden Fields ==
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!
!||||2 sin(𝝅/n)
!||||2 cos(𝝅/n)
!||||
!||||
!||||
!||||
!||||
!||||
!||||
|-
|1
|1
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|2
|&lt;small&gt;&lt;math&gt;\frac{\pi}{1}&lt;/math&gt;&lt;/small&gt;||180°||0.500
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|3
|_||120°||0.577
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|4
|&lt;small&gt;&lt;math&gt;\frac{\pi}{2}&lt;/math&gt;&lt;/small&gt;||90°||0.707
|2||90°||0.707
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|5
|_||144°||1.376
|𝝓||144°||1.376
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|6
|&lt;small&gt;&lt;math&gt;\frac{\pi}{3}&lt;/math&gt;&lt;/small&gt;||120°||1.732
|3||180°||2.000
|4||120°||1.732
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|7
|_||102.9°||2.077
|𝝆||154.3°||2.589
|𝝈||154.3°||2.589
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|8
|4₁||90°||2.414||4₂||135°||3.154||4₃||180°||3.414
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|9
|_||80°||2.747
|𝜶]||120°||3.702
|𝜷||160°||4.209
|𝜸||160°||4.209
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|10
|5₁||72°||3.078
|5₂||108°||4.236
|5₃||144°||4.980
|5₄||180°||5.236
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|11
|_||65.5°||3.406
|𝜽||98.2°||4.761
|𝜿||130.9°||5.730
|𝝀||163.6°||6.235
|𝝁||163.6°||6.235
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|12
|6₁||60°||3.732
|6₂||90°||5.278
|6₃||120°||6.464
|6₄||150°||7.210
|6₅||180°||7.464
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|13
|_||55.4°||4.057
|_||83.1°||5.789
|_||110.8°||7.185
|_||138.5°||8.163
|_.||166.2°||8.667
|_||166.2°||8.667
|_||138.5°||8.163
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|14
|7₁||51.4°||4.381
|7₂||77.1°||6.296
|7₃||102.9°||7.895
|7₄||128.6°||9.098
|7₅||154.3°||9.845
|7₆||180°||10.100
|7₇||154.3°||9.845
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|15
|_||48°||4.705
|_||72°||6.799
|_||96°||8.596
|_||120°||10.020
|_||144°||11.000
|_||168°||11.500
|_||168°||11.500
|_||144°||11.000
|_||120°||10.020
|}</text>
      <sha1>d1w7r394uzatvbv5n4rf2dm1xsgg45r</sha1>
    </revision>
    <revision>
      <id>2815454</id>
      <parentid>2815453</parentid>
      <timestamp>2026-06-12T23:33:41Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 600 cell */</comment>
      <origin>2815454</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="66681" sha1="014wjs308t0z6hi8c11blxpgyr5uqta" xml:space="preserve">== The 600 cell ==
[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} &lt;small&gt;&lt;math&gt;r_9=\phi&lt;/math&gt;&lt;/small&gt;]][Four {30/9}=3{10/3} over &lt;math&gt;r_9=\phi&lt;/math&gt; chords in the illustration is a distinct rotation arising in the 600-cell, one we shouldn't be illustrating here, unless we're going to illustrate all the non-edge 24-cell and 600-cell rotations.]

{{Clear}}

== Radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; 120-cell ==
{| 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|roots
!colspan=7|Chord lengths of the &lt;math&gt;\sqrt{2}&lt;/math&gt; 120-cell
|-
!colspan=5|length &lt;math&gt;c_t&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;\sqrt{2}&lt;/math&gt;
!colspan=2|length &lt;math&gt;c_t \times \phi^2/\sqrt{2}&lt;/math&gt;&lt;br&gt;in 120-cell of edge &lt;math&gt;1/\sqrt{2}&lt;/math&gt;, radius &lt;math&gt;c_8=\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,2}&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;\{30\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,2}-c_{2,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(3-\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.381966&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.145898}&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_{2,2}&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;\{\frac{30}{2}\}&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 \{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,2}-c_{4,2}\right)&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;\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_{3,2}&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;\{\frac{30}{3}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \{\frac{10}{3}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5}-1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.874032&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{2}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.763932}&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_{4,2}&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;\{\frac{60}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,2}}{\sqrt{2}}&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;\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_{5,2}&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;\{\frac{30}{4}\}&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 \{\frac{15}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{3} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.07047&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.1459}&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_{6,2}&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;\{\frac{120}{17}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,2}&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;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{\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[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_{7,2}&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;\{\frac{20}{3}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\frac{4}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.32813&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.76393}&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_{8,2}&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;\{\frac{30}{5}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{6\}&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;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;\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_{9,2}&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;\{\frac{40}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.54336&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.38197}&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_{10,2}&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;\{\frac{60}{11}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,2}&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{\phi ^2}&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;\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_{11,2}&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;\{\frac{30}{6}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{2 \sqrt[4]{5}}{\sqrt{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.66251&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 (3-\phi )}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.76393}&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_{12,2}&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;\{\frac{24}{5}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,2}&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{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_{13,2}&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;\{\frac{60}{13}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.83901&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.38197}&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_{14,2}&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;\{\frac{40}{9}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,2}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{1+\sqrt{5}}}{\sqrt{2}}&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[4]{5} \sqrt{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\sqrt{5} \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;\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_{15,2}&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;\{\frac{30}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,2}&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;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 \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_{16,2}&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;\{\frac{120}{29}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.09331&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.38197}&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_{17,2}&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;\{\frac{120}{31}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.14896&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.61803}&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_{18,2}&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;\{\frac{30}{8}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{15}{4}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.23607&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{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_{19,2}&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;\{\frac{30}{9}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{10}{3}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,2}+c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{\sqrt{2}}&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;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 (1+\phi )}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.23607}&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_{20,2}&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;\{\frac{120}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.31991&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.38197}&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_{21,2}&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;\{\frac{60}{19}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.37024&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 \left(\frac{5}{2}+\frac{1}{1+\sqrt{5}}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 \left(\frac{5}{2}+\frac{1}{1+\sqrt{5}}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.61803}&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_{22,2}&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;\{\frac{30}{10}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.44949&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.}&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_{23,2}&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;\{\frac{120}{41}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5+\frac{4}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.49721&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 \left(4-\frac{\psi }{2 \phi }\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.23607}&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_{24,2}&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;\{\frac{20}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.57255&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.61803}&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_{25,2}&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;\{\frac{30}{11}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{30}{11}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(7+3 \sqrt{5}\right)}&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;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.8541}&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_{26,2}&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;\{\frac{12}{5}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.64575&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.}&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_{27,2}&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;\{\frac{30}{12}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.68999&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2 (2+\phi )}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.23607}&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_{28,2}&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;\{\frac{30}{13}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{30}{13}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.76008&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.61803}&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_{29,2}&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;\{\frac{30}{14}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\frac{15}{7}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{3} \left(1+\sqrt{5}\right)&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;\sqrt{3} \phi &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;\sqrt{7.8541}&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_{30,2}&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;\{\frac{30}{15}\}&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 c_{8,2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.82843&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.}&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;
|}

The bitruncated {30/8} chord of the 120-cell provides a geometric derivation of the golden ratio formulas. Consider a 120-cell of radius &lt;small&gt;&lt;math&gt;2\sqrt{2}&lt;/math&gt;&lt;/small&gt; in which the {30/8} chord is &lt;small&gt;&lt;math&gt;2\sqrt{5}&lt;/math&gt;&lt;/small&gt; and the center section of the chord is &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;. Divide results by &lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt; to get a radius &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; result. The left section of the chord is:
:&lt;small&gt;&lt;math&gt;\tfrac{\sqrt{5} - 1}{2} \approx 0.618&lt;/math&gt;&lt;/small&gt;
The center section plus the right section is:
:&lt;small&gt;&lt;math&gt;\tfrac{1 + \sqrt{5}}{2} \approx 1.618&lt;/math&gt;&lt;/small&gt;
The sum of these two golden sections is &lt;small&gt;&lt;math&gt;\sqrt{5} \approx 2.236&lt;/math&gt;&lt;/small&gt;, the chord length.

== Radius &lt;math&gt;\phi&lt;/math&gt; 120-cell ==
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=9|Chord lengths of the &lt;math&gt;\phi&lt;/math&gt; 120-cell
|-
!&lt;math&gt;c_t&lt;/math&gt;
!arc
!&lt;math&gt;\frac{k}{d}&lt;/math&gt;
!colspan=4|length &lt;math&gt;c_t&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;\phi&lt;/math&gt;
!colspan=2|length &lt;math&gt;c_t\sqrt{2}&lt;/math&gt;&lt;br&gt;in 120-cell of edge &lt;math&gt;1/\phi&lt;/math&gt;, radius &lt;math&gt;c_8=\sqrt{2}\phi&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_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;30&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;\sqrt{\frac{2}{\left(1+\sqrt{5}\right)^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;\frac{1}{\phi }&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;c_2&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;15&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;\sqrt{\frac{1}{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;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_3&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;10&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;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{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;c_4&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;\frac{60}{7}&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;\sqrt{\frac{1}{8} \left(1+\sqrt{5}\right)^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;\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_5&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;\frac{15}{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{\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{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;c_6&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;\frac{120}{17}&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{\frac{1}{4} \sqrt{5} \left(1+\sqrt{5}\right)}&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{2} \sqrt{\frac{1}{\phi }}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi ^3}}{\phi }&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;c_7&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;\frac{20}{3}&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{\frac{1}{8} \left(1+\sqrt{5}\right) \left(-1+3 \sqrt{5}\right)}&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{\phi  \sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\psi  \phi ^3}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.14896&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_8&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;6&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{\frac{1}{4} \left(1+\sqrt{5}\right)^2}&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{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_9&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;\frac{40}{7}&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{\frac{1}{8} \left(1+\sqrt{5}\right) \left(1+3 \sqrt{5}\right)}&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;\frac{\phi  \sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^3}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.49721&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{10}&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;\frac{60}{11}&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;\sqrt{\frac{1}{32} \left(1+\sqrt{5}\right)^4}&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;\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_{11}&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;5&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;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(3+\frac{1}{2} \left(-1-\sqrt{5}\right)\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{3-\phi } \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.68999&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{12}&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;\frac{24}{5}&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{\frac{3}{8} \left(1+\sqrt{5}\right)^2}&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{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_{13}&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;\frac{60}{13}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{16} \left(9-\sqrt{5}\right) \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.10406&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.13657&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{14}&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;\frac{40}{9}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.73607}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{16} \sqrt{5} \left(1+\sqrt{5}\right)^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;\sqrt[4]{5} \phi ^{3/2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \phi  \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;8.05748&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{15}&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;\frac{30}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.23607}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)^2}&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;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.23607&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{16}&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;\frac{120}{29}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.73607}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{16} \left(11-\sqrt{5}\right) \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.39501&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.38705&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{17}&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;\frac{120}{31}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.04508}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4+\frac{1}{4} \left(-9+\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.45868&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4+\frac{1}{4} \left(\sqrt{5}-9\right)} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\psi  \phi ^5}}{\phi }&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_{18}&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;\frac{15}{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.54508}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{8} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.55834&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5} \sqrt{\phi ^4}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.61803&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{19}&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;\frac{10}{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.8541}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{16} \left(1+\sqrt{5}\right)^4}&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;\phi ^2&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_{20}&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;\frac{120}{37}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.04508}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4-\frac{1}{8} \left(1+\sqrt{5}\right)^2\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.65426&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{4-\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.75369&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{21}&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;\frac{60}{19}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.3541}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4-\frac{1+3 \sqrt{5}}{2 \left(1+\sqrt{5}\right)}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.71184&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{4-\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{8-\frac{\phi }{\chi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.45479&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{22}&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;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.8541}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{4} \left(1+\sqrt{5}\right)^2}&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;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.96336&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{23}&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;\frac{120}{41}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.16312}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4-\frac{-1+3 \sqrt{5}}{2 \left(1+\sqrt{5}\right)}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.85712&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^5}}{\phi }&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_{24}&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;\frac{20}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.66312}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4-\frac{\sqrt{5}}{1+\sqrt{5}}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.94332&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\sqrt{5}}{2 \phi }} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.16248&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{25}&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;\frac{30}{11}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.97214}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{128} \left(1+\sqrt{5}\right)^6}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.99535&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^3}{\sqrt{2}}&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_{26}&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;\frac{12}{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.16312}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{8} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.02706&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.28092&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{27}&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;\frac{5}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.47214}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(2+\frac{1}{2} \left(1+\sqrt{5}\right)\right)}&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;\phi  \sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{2 \phi +4}&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_{28}&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;\frac{30}{13}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.97214}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(1+\sqrt{5}\right)^2 \left(4-\frac{2}{\left(1+\sqrt{5}\right)^2}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.15787&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{1}{2 \phi ^2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.4659&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{29}&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;\frac{15}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.2812}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{32} \left(1+\sqrt{5}\right)^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.20642&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{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_{30}&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;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.4721}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.23607&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.57649&lt;/math&gt;&lt;/small&gt;
|}

== Radius &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; 120-cell ==
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=9|Chord lengths of the &lt;math&gt;\sqrt{3}&lt;/math&gt; 120-cell
|-
!&lt;math&gt;c_t&lt;/math&gt;
!arc
!&lt;math&gt;\frac{k}{d}&lt;/math&gt;
!colspan=4|length &lt;math&gt;c_t&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;\sqrt{3}&lt;/math&gt;
!colspan=2|length &lt;math&gt;c_t \times c_8/\sqrt{3}&lt;/math&gt;&lt;br&gt;in 120-cell of edge &lt;math&gt;1/\sqrt{3}&lt;/math&gt;, radius &lt;math&gt;c_8=\sqrt{\frac{2}{3}}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_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;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.218847}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{24}{\left(1+\sqrt{5}\right)^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.467811&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.57735&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_2&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;15&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{\frac{6}{\left(1+\sqrt{5}\right)^2}}&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;\frac{\phi }{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.934172&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_3&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;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.1459}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{12}{\left(1+\sqrt{5}\right)^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.07047&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{3}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.32112&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_4&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;\frac{60}{7}&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{\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;\frac{\phi ^2}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51152&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_5&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;\frac{15}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.71885}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{18}{\left(1+\sqrt{5}\right)^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.31105&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{3}{\sqrt{2} \phi }&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_6&lt;/math&gt;&lt;/small&gt;
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|-
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|-
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|-
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|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^3}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
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|-
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|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{8} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;2.44569&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{11}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{3}} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.51292&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{12}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt; 
|-
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|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{6} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\phi ^2}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
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|-
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|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\frac{\phi }{\chi }}}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
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|-
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|-
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|&lt;small&gt;&lt;math&gt;\frac{\phi ^4}{\sqrt{3}}&lt;/math&gt;&lt;/small&gt;
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|-
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|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{3}} \phi ^2&lt;/math&gt;&lt;/small&gt;
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|-
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|-
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|-
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|-
|&lt;small&gt;&lt;math&gt;c_{30}&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;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{12.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{12}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.4641&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{\frac{2}{3}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.27523&lt;/math&gt;&lt;/small&gt;
|}

== Radius &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt; 120-cell ==
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=9|Chord lengths of the &lt;math&gt;\sqrt{5}&lt;/math&gt; 120-cell
|-
!&lt;math&gt;c_t&lt;/math&gt;
!arc
!&lt;math&gt;\frac{k}{d}&lt;/math&gt;
!colspan=4|length &lt;math&gt;c_t&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;\sqrt{5}&lt;/math&gt;
!colspan=2|length &lt;math&gt;c_t \times c_8/\sqrt{5}&lt;/math&gt;&lt;br&gt;in 120-cell of edge &lt;math&gt;1/\sqrt{5}&lt;/math&gt;, radius &lt;math&gt;c_8 = \sqrt{\frac{2}{5}} \phi ^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_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;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.364745}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{40}{\left(1+\sqrt{5}\right)^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.603941&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{5}{2}}}{\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.447214&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_2&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;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.954915}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{10}{\left(1+\sqrt{5}\right)^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.977198&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{5}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.723607&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_3&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;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.90983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{20}{\left(1+\sqrt{5}\right)^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.38197&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{5}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.02333&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_4&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;\frac{60}{7}&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{\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;\frac{\phi ^2}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17082&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_5&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;\frac{15}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.86475}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{30}{\left(1+\sqrt{5}\right)^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.69256&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{15}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{5}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.25332&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_6&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;\frac{120}{17}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.45492}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5 \sqrt{5}}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85874&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{5^{3/4} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\phi ^3}}{\sqrt[4]{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.37638&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_7&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;\frac{20}{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.40983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5 \left(-1+3 \sqrt{5}\right)}{2 \left(1+\sqrt{5}\right)}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.09996&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} \sqrt{\frac{\psi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\psi  \phi ^3}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.555&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_8&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;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.23607&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{5}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.65579&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_9&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;\frac{40}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5.95492}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5 \left(1+3 \sqrt{5}\right)}{2 \left(1+\sqrt{5}\right)}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.44027&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} \sqrt{\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^3}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.807&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{10}&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;\frac{60}{11}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.54508}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{8} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.55834&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^3}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.89443&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{11}&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;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6.90983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(3+\frac{1}{2} \left(-1-\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.62866&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{5}} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.9465&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{12}&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;\frac{24}{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{15}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.73861&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{15}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{5}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.02792&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{13}&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;\frac{60}{13}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{8.45492}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.90773&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5 \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{10} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.822431&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{14}&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;\frac{40}{9}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{9.04508}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{4} \sqrt{5} \left(1+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.25325&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5^{3/4} \sqrt{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{\phi ^5}}{\sqrt[4]{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.83045&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{15}&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;\frac{30}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.16228&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{2 \phi ^2}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.34164&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{16}&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;\frac{120}{29}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{10.9549}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.30982&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5 \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{10} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.4509&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{17}&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;\frac{120}{31}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{11.5451}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4+\frac{1}{4} \left(-9+\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.39781&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4+\frac{1}{4} \left(\sqrt{5}-9\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\psi  \phi ^5}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.51605&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{18}&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;\frac{15}{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{12.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{25}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.53553&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{5}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi ^4}&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_{19}&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;\frac{10}{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{13.0902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{4} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{2}{5}} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.67912&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{20}&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;\frac{120}{37}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{13.4549}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4-\frac{1}{8} \left(1+\sqrt{5}\right)^2\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.66809&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{4-\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\phi ^2}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.71619&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{21}&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;\frac{60}{19}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{14.0451}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4-\frac{1+3 \sqrt{5}}{2 \left(1+\sqrt{5}\right)}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.74768&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{4-\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\frac{\phi }{\chi }}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.22352&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{22}&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;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{15.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{15}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.87298&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{15}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{6}{5}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.86791&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{23}&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;\frac{120}{41}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{15.5902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4-\frac{-1+3 \sqrt{5}}{2 \left(1+\sqrt{5}\right)}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.94844&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\chi  \phi ^5}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.92379&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{24}&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;\frac{20}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{16.5451}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4-\frac{\sqrt{5}}{1+\sqrt{5}}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.06756&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{4-\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.012&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{25}&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;\frac{30}{11}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{17.1353}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{32} \left(1+\sqrt{5}\right)^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.13948&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^4}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.06525&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{26}&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;\frac{12}{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{17.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{35}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.1833&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{35}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{5}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.0977&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{27}&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;\frac{5}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{18.0902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(2+\frac{1}{2} \left(1+\sqrt{5}\right)\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.25325&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{2 \phi +4}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.1495&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{28}&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;\frac{30}{13}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{19.0451}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5 \left(4-\frac{2}{\left(1+\sqrt{5}\right)^2}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.36407&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{4-\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}}{\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.23156&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{29}&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;\frac{15}{7}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{19.6353}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{15}{8} \left(1+\sqrt{5}\right)^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.43117&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{15}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{5}} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.28124&lt;/math&gt;&lt;/small&gt; 
|-
|&lt;small&gt;&lt;math&gt;c_{30}&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;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{20.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{20}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.47214&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{\frac{2}{5}} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.31158&lt;/math&gt;&lt;/small&gt;
|}

== Steinbach's golden chords ==
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=11|
|-
!
!
!colspan=2 align=left|pentagon {5}
!colspan=2 align=left|heptagon {7}
!colspan=2 align=left|nonagon {9}
!colspan=2 align=left|hendecagon {11}
!
|-
|
|
|colspan=2 align=left|
&lt;small&gt;&lt;math&gt;\phi=\frac{1}{2}(1 + \sqrt{5}) \approx 1.618034&lt;/math&gt;&lt;/small&gt;
|colspan=2 align=left|
&lt;small&gt;&lt;math&gt;\rho=2\cos{\pi/7} \approx 1.80194&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\sigma=\rho^2 - 1 \approx 2.24698&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
|colspan=2 align=left|
&lt;small&gt;&lt;math&gt;\alpha=2\cos{\pi/9} \approx 1.87939&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\beta=&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\gamma=&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
|colspan=2 align=left|
&lt;small&gt;&lt;math&gt;\theta=2\cos{\pi/11} \approx 1.91899&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\kappa=&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\lambda=&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\mu=&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
|
|}

{{Efn|&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi=\frac{1}{2}(\sqrt{5} + 1) \approx 1.618034&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\Phi=\frac{1}{2}(\sqrt{5} - 1) \approx 0.618034&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\chi=\frac{1}{2}(3\sqrt{5} + 1) \approx 3.854102&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\psi=\frac{1}{2}(3\sqrt{5} - 1) \approx 2.854102&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
|name=phi constants}}

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=20|Chord lengths of the 120-cell
|-
|1
|1.
|-                        
|2
|1.
|-                        
|3
|1.
|-                       
|4
|1.
|2
|90°
|1.00
|-                     
|5
|1.
|&lt;math&gt;\phi&lt;/math&gt;
|144°
|1.62
|-                     
|6
|1.
|3
|120°
|1.73
|4
|180°
|2.00 
|-                 
|7
|1.
|&lt;math&gt;\rho&lt;/math&gt;
|102.9°
|1.80
|&lt;math&gt;\sigma&lt;/math&gt;
|154.3°
|2.25
|}
{{Void|                
8	1.	4₂	90 °	1.85	4₃	135 °	2.41                  
9	1.	&lt;math&gt;alpha&lt;/math&gt;	80 °	1.88	&lt;math&gt;beta&lt;/math&gt;	120 °	2.53	&lt;math&gt;gamma&lt;/math&gt;	160 °	2.88               
10	1.	5₂	72 °	1.90	5₃	108 °	2.62	5₄	144 °	3.08               
11	1.	&lt;math&gt;theta&lt;/math&gt;	65.5 °	1.92	&lt;math&gt;kappa&lt;/math&gt;	98.2 °	2.68	&lt;math&gt;lambda&lt;/math&gt;	130.9 °	3.23	&lt;math&gt;mu&lt;/math&gt;	163.6 °	3.51            
12	1.	6₂	60 °	1.93	6₃	90 °	2.73	6₄	120 °	3.35	6₅	150 °	3.73            
13	1.	_	55.4 °	1.94	_	83.1 °	2.77	_	110.8 °	3.44	_	138.5 °	3.91	_	166.2 °	4.15	_	166.2 °	4.15      
14	1.	_₂	51.4 °	1.95	_₃	77.1 °	2.80	_₄	102.9 °	3.51	_₅	128.6 °	4.05	_₆	154.3 °	4.38	_₇	180 °	4.49      
15	1.	_	48 °	1.96	_	72 °	2.83	_	96 °	3.57	_	120 °	4.17	_	144 °	4.57	_	168 °	4.78	_	168 °	4.78	_	144 °	4.57
16	1.	_₂	45 °	1.96	_₃	67.5 °	2.85	_₄	90 °	3.62	_₅	112.5 °	4.26	_₆	135 °	4.74	_₇	157.5 °	5.03	_₈	180 °	5.13	_₉	157.5 °	5.03
}}
{{Void|
chordL :={
{{&quot;1₂&quot;}},
{{&quot;1₁&quot;}},
{{&quot;3₂&quot;}},
{{&quot;2₁&quot;}},
{{&quot;5₂&quot;},{ &quot;5₄&quot;}},
{{&quot;3₁&quot;},{ &quot;3₂&quot;}},
{{&quot;7₂&quot;},{&quot;7₄&quot;},{&quot;7₆&quot;}},
{{&quot;4₁&quot;},{ &quot;4₂&quot;},{ &quot;4₃&quot;}},
{{&quot;9₂&quot;}, {&quot;9₄&quot;},{&quot;9₆&quot;},{&quot;9₈&quot;}},
{{&quot;5₁&quot;}, { &quot;5₂&quot;},{ &quot;5₃&quot;},{ &quot;5₄&quot;}},
{{&quot;11₂&quot;},{&quot;11₄&quot;},{&quot;11₆&quot;},{&quot;11₈&quot;},{&quot;11₁₀&quot;}} ,
{{&quot;6₁&quot;},{ &quot;6₂&quot;},{ &quot;6₃&quot;},{ &quot;6₄&quot;},{ &quot;6₅&quot;}},
{{&quot;13₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}, { &quot;_&quot;}},
{{&quot;7₁&quot;},{ &quot;7₂&quot;},{ &quot;7₃&quot;},{ &quot;7₄&quot;},{ &quot;7₅&quot;}, { &quot;7₆&quot;}},
{{&quot;15₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}, { &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;8₁&quot;},{ &quot;8₂&quot;},{ &quot;8₃&quot;},{ &quot;8₄&quot;},{ &quot;8₅&quot;},{ &quot;8₆&quot;},{ &quot;8₇&quot;}},
{{&quot;17₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;9₁&quot;},{ &quot;9₂&quot;},{ &quot;9₃&quot;},{ &quot;9₄&quot;},{ &quot;9₅&quot;},{ &quot;9₆&quot;},{ &quot;9₇&quot;},{ &quot;9₈&quot;}},
{{&quot;19₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;10₁&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;21₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;11₁&quot;},{ &quot;11₂&quot;},{ &quot;11₃&quot;},{ &quot;11₄&quot;},{ &quot;11₅&quot;},{ &quot;11₆&quot;},{ &quot;11₇&quot;},{ &quot;11₈&quot;},{ &quot;11₉&quot;},{ &quot;11₁₀&quot;}},
{{&quot;23₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;12₁&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;25₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;13₁&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;27₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;15₁&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;29₂&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}},
{{&quot;16₁&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;},{ &quot;_&quot;}}
}}

== Golden Fields ==
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!
!||||2 sin(𝝅/n)
!||||2 cos(𝝅/n)
!||||
!||||
!||||
!||||
!||||
!||||
!||||
|-
|1
|1
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|2
|&lt;small&gt;&lt;math&gt;\frac{\pi}{1}&lt;/math&gt;&lt;/small&gt;||180°||0.500
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|3
|_||120°||0.577
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|4
|&lt;small&gt;&lt;math&gt;\frac{\pi}{2}&lt;/math&gt;&lt;/small&gt;||90°||0.707
|2||90°||0.707
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|5
|_||144°||1.376
|𝝓||144°||1.376
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|6
|&lt;small&gt;&lt;math&gt;\frac{\pi}{3}&lt;/math&gt;&lt;/small&gt;||120°||1.732
|3||180°||2.000
|4||120°||1.732
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|7
|_||102.9°||2.077
|𝝆||154.3°||2.589
|𝝈||154.3°||2.589
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|8
|4₁||90°||2.414||4₂||135°||3.154||4₃||180°||3.414
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|9
|_||80°||2.747
|𝜶]||120°||3.702
|𝜷||160°||4.209
|𝜸||160°||4.209
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|10
|5₁||72°||3.078
|5₂||108°||4.236
|5₃||144°||4.980
|5₄||180°||5.236
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|11
|_||65.5°||3.406
|𝜽||98.2°||4.761
|𝜿||130.9°||5.730
|𝝀||163.6°||6.235
|𝝁||163.6°||6.235
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|12
|6₁||60°||3.732
|6₂||90°||5.278
|6₃||120°||6.464
|6₄||150°||7.210
|6₅||180°||7.464
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|13
|_||55.4°||4.057
|_||83.1°||5.789
|_||110.8°||7.185
|_||138.5°||8.163
|_.||166.2°||8.667
|_||166.2°||8.667
|_||138.5°||8.163
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|14
|7₁||51.4°||4.381
|7₂||77.1°||6.296
|7₃||102.9°||7.895
|7₄||128.6°||9.098
|7₅||154.3°||9.845
|7₆||180°||10.100
|7₇||154.3°||9.845
|- style=&quot;text-align:right&quot;
| style=&quot;text-align:left&quot;|15
|_||48°||4.705
|_||72°||6.799
|_||96°||8.596
|_||120°||10.020
|_||144°||11.000
|_||168°||11.500
|_||168°||11.500
|_||144°||11.000
|_||120°||10.020
|}</text>
      <sha1>014wjs308t0z6hi8c11blxpgyr5uqta</sha1>
    </revision>
  </page>
  <page>
    <title>The Ignorant Observer Framework</title>
    <ns>0</ns>
    <id>329703</id>
    <revision>
      <id>2815473</id>
      <parentid>2812630</parentid>
      <timestamp>2026-06-13T10:12:32Z</timestamp>
      <contributor>
        <username>IgnorantObserver</username>
        <id>3076980</id>
      </contributor>
      <comment>Major repositioning: state explicitly that the strong BLQC reading (unrecoverable visibility loss) is excluded by existing experiments; reframe IOF as a deterministic no-collapse interpretation carrying a recoverable, calibrated control-theoretic subtheory; rename source documents (Where Did the Measurement Basis Come From, Born Rule from Finite Observation, Capacity-Calibrated Penrose test); restructure lead with three-layer 'What kind of claim this is' taxonomy; add four references (Bartlet...</comment>
      <origin>2815473</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="39454" sha1="i09ett2lmaj6qtahslfiby53l7xw341" xml:space="preserve">{{Research project}}

= The Ignorant Observer Framework =

''This research page is authored and maintained by [[User:IgnorantObserver|Aernoud Dekker]], an independent researcher and the originator of the framework described below. Page text is offered for review, critique, and collaborative refinement under [[Wikiversity:Copyrights|Wikiversity's standard licence]].''

== Status ==

Research project under active development. The framework is an interlinked set of technical and interpretive documents published at [https://ignorantobserver.xyz ignorantobserver.xyz] and archived on the [https://osf.io Open Science Framework]. ''The Ignorant Observer'' is the foundational paper. ''Where Did the Measurement Basis Come From?'' is the conceptual bridge that states, for a physics reader, exactly what claim the framework makes about the measurement basis. ''Bandwidth-Limited Quantum Control'' (BLQC) is the operational layer: a finite-rate phase-reference control law and its laboratory benchmark. ''A Capacity-Calibrated Protocol for Testing Penrose Objective Reduction'' applies that benchmark as the calibration arm of a test of gravitational collapse. A companion paper, ''The Born Rule from Finite Observation'', derives the binary Born form conditionally from finite-record geometry, and a no-go companion marks where that route stops. All work is single-authored.

'''Where the framework stands (June 2026).''' IOF is an interpretation of quantum mechanics, not a rival to it. Its central empirical claim was once sharpened to a testable form — that finite basis tracking produces an ''unrecoverable'' visibility loss, a genuine deviation from standard quantum mechanics — and that strong reading was found to be '''excluded by existing experiments'''. What survives, and what the framework now is, is a deterministic, no-collapse ''interpretation'' carrying a ''recoverable'', calibrated control-theoretic subtheory. This page is written in that developed position: it raises the strong reading and states its exclusion rather than asserting it.

== One-sentence thesis ==

Quantum measurement normally treats the basis as if it came from outside physics. The Ignorant Observer Framework treats the basis as a finite physical reference variable generated and tracked inside the apparatus itself — and reads quantum probability as the form a single deterministic history takes in the records of an observer that cannot fully resolve that reference.

== What kind of claim this is ==

The framework has three layers, kept deliberately in separate books and adjudicated by different standards. Confusing them is the most common misreading, so they are named up front.

* '''Operational layer''' (classical, measurable). The deficit κ = ''h''&lt;sub&gt;KS&lt;/sub&gt; − ''C''&lt;sub&gt;eff&lt;/sub&gt; ln 2 is a calibrated control law for the visibility of ''unconditioned'' records. It is real physics — but it is standard quantum mechanics plus classical control theory, so confirming it '''benchmarks the apparatus; it does not discriminate IOF from quantum mechanics'''. Its hallmark is ''recoverability'': the lost contrast returns when the realised reference is supplied.
* '''Foundations layer''' (structural reconstruction). A conditional derivation obtains the binary Born weight ''p''(θ) = cos²(θ/2) from finite-record geometry under two named bridge assumptions; a companion no-go theorem shows the same route cannot reach quantum phase or the multi-outcome rule. This is a reconstruction, not an independent empirical prediction — a constant Fisher information ''I''(θ) is itself a standard-quantum prediction, so the test of the bridge premise is a consistency check.
* '''Interpretation layer''' (the framework's identity). A single deterministic global history, sampled by finite embedded observers; apparent collapse is the record-relative update of a bounded observer, and quantum randomness is the form taken by a dependency the observer cannot resolve. As an interpretation it is a peer of many-worlds, Bohmian mechanics, QBism, and relational quantum mechanics, and is judged — as they are — by coherence and economy, not by a discriminating experiment.

Because the framework as a whole reproduces standard quantum mechanics, it is '''not falsifiable as a whole — by design, not by evasion'''. Its falsifiable content is modular and named: the operational control law (with the caveat that a failure there indicts the apparatus model and the calibration of ''C''&lt;sub&gt;eff&lt;/sub&gt; and ''h''&lt;sub&gt;KS&lt;/sub&gt; before it indicts the framework); the Fisher-homogeneity premise of the Born bridge (exposed in the falsifying direction only); and a registered chaotic-corner module carried with the prior strongly against it.

== Core question ==

''When the classical degrees of freedom that define a measurement basis generate information faster than the apparatus can track them, do the apparatus's raw, unconditioned records lose interference contrast on a schedule set by the deficit'' κ = ''h''&lt;sub&gt;KS&lt;/sub&gt; − ''C''&lt;sub&gt;eff&lt;/sub&gt; ln 2 ''— and is that loss recoverable by conditioning on a log of the realised reference?''

The framework's answer is yes to both, and the second half is the point. The loss is reference-frame physics ''within'' quantum mechanics, not a new channel beyond it: standard quantum mechanics already says that records taken against a blurred reference lose contrast, and recovers that contrast once the reference is restored. IOF's contribution is to make the blur a calibrated, controllable quantity — a number a laboratory can dial up and down.

== The strong reading, and why it was excluded ==

A bolder reading of the same formula would make the visibility loss ''unrecoverable'': a capacity-dependent suppression that no offline log could undo — that is, a measurable deviation from standard quantum mechanics. Sharpened, that reading requires the unresolved basis not to imprint on the records (a Markov condition giving a recovery statistic ''R''&lt;sub&gt;rec&lt;/sub&gt; ≈ 0).

Existing experiments exclude it. Logged-setting Bell tests recover the full quantum correlation when the realised settings are recorded and the data are sorted offline (Weihs et al. 1998); randomized-measurement tomography and classical shadows choose observables ''after'' measurement and reconstruct the quantum state (Huang, Kueng &amp; Preskill 2020); correlation spectroscopy recovers coherence beyond the laser linewidth from cross-records. In every case the lost contrast comes back — ''R''&lt;sub&gt;rec&lt;/sub&gt; ≈ 1 up to ordinary imperfections. The suppression is phase averaging over an unresolved reference, not a new physical channel; IOF's own diffusive formula correctly describes the ''raw, unconditioned'' signal, and the recovery is what reveals it as bookkeeping. The framework therefore raises the strong reading and excludes it; it does not claim it.

One corner remains formally open, and is registered rather than asserted: recoverability under ''certified deterministic-chaotic'' basis dynamics has not been directly tested. It is carried in the protocol with the prior strongly against any difference from the diffusive case, and a positive there would overturn the corpus's own exclusion, not confirm a surviving claim.

== Technical proposal ==

The framework introduces the following quantities.

'''Effective channel capacity ''C''&lt;sub&gt;eff&lt;/sub&gt;''' (bits/s): the information rate available to the basis-tracking control loop, operationalised as
:''C''&lt;sub&gt;eff&lt;/sub&gt; = ''r'' · ''b'' · ''f''
with ''r'' the update rate (Hz), ''b'' the effective bits per update that constrain the basis variable θ, and ''f'' ∈ (0,1] the fraction of updates that genuinely constrain θ after overhead and latency. ''C''&lt;sub&gt;eff&lt;/sub&gt; is ultimately bounded by the Landauer limit on the controller's irreversible bookkeeping,
:''C''&lt;sub&gt;eff&lt;/sub&gt; ≤ ''P'' / (''k''&lt;sub&gt;B&lt;/sub&gt; ''T'' ln 2),
but this is a thermodynamic ceiling, not the operating point: realised ''C''&lt;sub&gt;eff&lt;/sub&gt; is typically far lower and must be calibrated as ''useful'' information actually constraining θ, since added power can equally couple to actuator noise or backaction channels that do not constrain the basis.

'''Kolmogorov–Sinai entropy rate ''h''&lt;sub&gt;KS&lt;/sub&gt;''' (nats/s): the information-production rate of the classical degrees of freedom (voltage references, timing circuits, feedback loops) that define and maintain the measurement basis. For chaotic systems ''h''&lt;sub&gt;KS&lt;/sub&gt; equals the sum of positive Lyapunov exponents (Pesin identity); it is estimated from the exponential growth of one-step prediction error on logged controller states. The nats/s convention lets the deficit κ combine ''h''&lt;sub&gt;KS&lt;/sub&gt; (nats/s) and ''C''&lt;sub&gt;eff&lt;/sub&gt; ln 2 (bits/s converted to nats/s) in consistent units.

'''Self-ignorance rate κ''' (s&lt;sup&gt;−1&lt;/sup&gt;):
:κ = ''h''&lt;sub&gt;KS&lt;/sub&gt; − ''C''&lt;sub&gt;eff&lt;/sub&gt; · ln 2
with two regimes. When κ &lt; 0 (''capacity-wins''), basis-tracking error stays bounded and standard quantum visibility is recovered, modulo ordinary decoherence. When κ &gt; 0 (''chaos-wins''), '''standard quantum mechanics still holds''' — nothing new happens to the world — but the observer's reference is no longer fully resolved, and the variance of the basis-tracking error grows as σ&lt;sub&gt;θ&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;(''t'') = σ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt; e&lt;sup&gt;2κ''t''&lt;/sup&gt;, so the ''raw, unconditioned'' records lose contrast.

'''Measured visibility ''V''(''t'')'''. Averaging the interference term cos(φ − θ) over a Gaussian basis-tracking error δθ ∼ ''N''(0, σ&lt;sub&gt;θ&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;(''t'')) yields, in the small-angle regime,
:''V''(''t'') = exp(−½ σ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt; e&lt;sup&gt;2κ''t''&lt;/sup&gt;),
a ''double-exponential'' decay of the unconditioned visibility once the chaos-wins regime is entered.

'''Two visibility channels (epistemic optics)'''. The basis-tracking loss is one of two multiplicative contributions to the observed visibility:
:''V''&lt;sub&gt;obs&lt;/sub&gt; = ''V''&lt;sub&gt;std&lt;/sub&gt; · ''V''&lt;sub&gt;IOF&lt;/sub&gt;,    with    ''V''&lt;sub&gt;IOF&lt;/sub&gt; = exp(−½ σ&lt;sub&gt;θ&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;),
under the Gaussian independent-smearing model, where ''V''&lt;sub&gt;std&lt;/sub&gt; is the ordinary physical visibility standard quantum mechanics already predicts and ''V''&lt;sub&gt;IOF&lt;/sub&gt; is the ''accessible-reference'' visibility — the contrast available to a finite record. The framework does not deny ''V''&lt;sub&gt;std&lt;/sub&gt;; it claims that, in the chaos-wins regime, part of the contrast lost from the raw record belongs to ''V''&lt;sub&gt;IOF&lt;/sub&gt; and is recoverable, and may be misassigned to standard decoherence if the capacity–instability coordinate κ is not independently varied, conditioned, and tested.

'''Breakdown time ''t''&lt;sub&gt;break&lt;/sub&gt;'''. For a chosen visibility threshold ''V''*,
:''t''&lt;sub&gt;break&lt;/sub&gt; = (1 / 2κ) · ln(−2 ln ''V''* / σ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;),    κ &gt; 0.
''t''&lt;sub&gt;break&lt;/sub&gt; is the operational layer's primary observable.

The derivation extends the Data-Rate Theorem of Nair &amp; Evans (2004) and Tatikonda &amp; Mitter (2004) from linear plants to nonlinear, chaotic systems by substituting ''h''&lt;sub&gt;KS&lt;/sub&gt; for the sum-of-positive-eigenvalues bound. This extension is an explicit assumption of the framework, not a proven theorem (see [[#Open review targets|Open review targets]]).

== The operational benchmark and the Penrose test ==

The experimental layer has two roles, and keeping them apart is essential: an operational ''benchmark'' that calibrates the apparatus, and a ''discrimination'' arm that tests gravitational collapse. Neither tests IOF against quantum mechanics.

'''The benchmark (calibration arm).''' Vary the useful basis-tracking capacity ''C''&lt;sub&gt;eff&lt;/sub&gt; — through the accepted update rate, useful bit depth, estimator bandwidth, or a calibrated packet-drop schedule — at clamped temperature, readout signal-to-noise, latency, actuator behaviour, and mass geometry. The prediction is that the raw-record breakdown time ''t''&lt;sub&gt;break&lt;/sub&gt; moves with κ = ''h''&lt;sub&gt;KS&lt;/sub&gt; − ''C''&lt;sub&gt;eff&lt;/sub&gt; ln 2, with the decay carrying the double-exponential form rather than a single exponential or Gaussian. This is a genuine, pre-registered test of the control law — but its confirmation is ''expected'': standard quantum mechanics plus finite-rate control already predict it. A positive result therefore demonstrates per-shot command of ''C''&lt;sub&gt;eff&lt;/sub&gt; and ''h''&lt;sub&gt;KS&lt;/sub&gt; — it benchmarks the instrument — and a failure indicts the apparatus model or the calibration before it indicts the framework. It does not discriminate IOF from quantum mechanics, because nothing in it lies beyond quantum mechanics.

'''The recoverability classifier.''' The decisive operational quantity is the recovery statistic ''R''&lt;sub&gt;rec&lt;/sub&gt;: condition the raw records on a passive, full-resolution shadow log of the realised reference and ask whether the lost contrast returns. ''Recoverable'' (''R''&lt;sub&gt;rec&lt;/sub&gt; → 1) is reference-frame physics — the framework's own expected case, in the lineage of quantum reference frames (Bartlett, Rudolph &amp; Spekkens 2007). ''Unrecoverable'' loss is decoherence-type or collapse-type. The classifier is what separates the framework's subject from everything that is not its subject.

'''The discrimination arm — a test of Penrose, not of IOF.''' With tracking held at maximal capacity and the engineered channel quiet, sweep mass geometry and look for a κ-independent, ''unrecoverable'' visibility floor — the signature of gravitational [[w:Penrose interpretation|objective reduction]] (OR), whose timescale ''τ''&lt;sub&gt;OR&lt;/sub&gt; ≈ ℏ/''E''&lt;sub&gt;G&lt;/sub&gt; is fixed by the gravitational self-energy of the superposition. The framework's contribution here is methodological: the calibrated reference channel and the ''R''&lt;sub&gt;rec&lt;/sub&gt; classifier measure and subtract everything the observer's own finitude contributes, so that a null becomes a genuine bound on objective reduction and a surviving floor is certified as nature's rather than the apparatus's. '''The experiment tests Penrose's proposal, not IOF against quantum mechanics''': if an unrecoverable floor is found, quantum mechanics itself needs repair and Penrose is vindicated; if none is found, his proposal is constrained — and IOF remains exactly what it was, a way of understanding why quantum mechanics works. Neither outcome discriminates IOF from quantum mechanics.

This discipline has a sharp boundary, worth stating: calibration can move only the probability of a ''correct verdict given the data''; it cannot move the probability that objective reduction is true. The instrument decides; it does not vote.

The architecture is also not specific to Penrose. The same calibrate–subtract–classify procedure reads against the whole dynamical-collapse family — spontaneous-localisation (GRW/CSL) models and Diósi–Penrose gravitational collapse alike — distinguished by the variable the surviving floor scales with (mass and geometry for OR, nucleon-number amplification for GRW/CSL). Penrose is the flagship target because its schedule lands in an experimentally accessible mesoscopic window and carries a live debate, not because the method is bespoke to it. The [https://www.qgemproject.com/ QGEM] pathfinder is one candidate testbed; superconducting-qubit readout chains and precision-interferometer phase-locking loops are others.

'''The Fisher-homogeneity module.''' A separate module tests the Born-derivation bridge by measuring whether the empirical Fisher information ''I''(θ) is approximately constant across the calibrated basis range, as the scalar-threshold-homogeneity premise requires. Because a flat ''I''(θ) is ''also'' what standard quantum mechanics predicts for an ideal binary readout, a homogeneous result confirms a shared prediction and lends no positive weight to the bridge; the module is a falsification-only check on a foundations premise, logically independent of the κ-benchmark.

== Relation to quantum foundations ==

* '''The measurement problem'''. The framework relocates it rather than patching it: nothing in the world makes the passage from superposition to a single record; the field |ψ⟩ goes on evolving unitarily, the underlying history stays definite, and what changes discontinuously is the observer's access — from a distribution over its own unresolved basis to a registered fact. &quot;Collapse&quot; is the name a bounded observer gives to updating its own ledger. The question is no longer &quot;how does an indeterminate world become definite?&quot; but &quot;why does a single definite history read, from the inside, as a probability?&quot; — and the answer is finite self-ignorance. Developed at length in ''[https://ignorantobserver.xyz/documents/Where_Did_the_Measurement_Basis_Come_From.pdf Where Did the Measurement Basis Come From?]''
* '''Indexed objectivity'''. The definiteness an observer meets is objective ''indexed to its configuration'' — irreducible at that access, found the same way by every inspector there. Penrose's proposal is precisely that the index can be dropped: that definiteness is absolute, gravitationally enforced, with no log anywhere able to restore the superposition. ''R''&lt;sub&gt;rec&lt;/sub&gt; is that question made operational — ''does the index drop?''
* '''ψ-ontic, collapse-epistemic'''. Within the empirical model the wavefunction is ontic and never collapses, so no-go results in the style of Pusey–Barrett–Rudolph — which constrain ψ-''epistemic'' models — do not bear against the framework; they support the wavefunction realism IOF requires. What is epistemic in IOF is ''collapse'': the record-relative update of a finite observer.
* '''Relational Quantum Mechanics''' (Rovelli) takes outcomes to be relative to an observer–system; the framework offers one candidate physical mechanism — finite ''C''&lt;sub&gt;eff&lt;/sub&gt; against ''h''&lt;sub&gt;KS&lt;/sub&gt; — for what makes one observer's frame physically inequivalent to another's.
* '''Decoherence theory''' is not opposed. The framework brought one further thing inside the physics that decoherence leaves outside: the causal history of the measurement ''setting''. In the capacity-wins regime standard decoherence theory is recovered.
* '''Measurement independence'''. Extended to Bell-type set-ups, IOF does not assume measurement independence: the setting and the system share a common past, so they need not be statistically independent. This places the framework in the superdeterministic family in the technical sense only — non-conspiratorial, with no fine-tuned ledger and no signalling, in the spirit of Palmer (2024). The shared ancestry is real but structurally inaccessible to the embedded observer; the framework names this ''epistemically bounded ancestral correlation''. IOF does not derive the Bell correlations from this alone — the Born weights and joint correlations are supplied by the hosted no-collapse embedding and recovered in the capacity-wins limit. A full consistency treatment, including a no-signalling lemma, remains an [[#Open review targets|open review target]].
* '''Information geometry'''. The conditional binary-Born derivation runs from finite-record constraints through a Fisher capacity bridge (identifying ''C''&lt;sub&gt;eff&lt;/sub&gt; with capacity for preserving operational distinguishability), Cencov's uniqueness theorem (selecting Fisher–Rao as the invariant distinguishability metric), and square-root record coordinates, to scalar-threshold homogeneity of κ in θ and thence ''p''(θ) = cos²(θ/2). The connection between statistical distance and quantum probability is not new (Wootters 1981); the framework runs the logic from finite-observer record geometry. A companion no-go theorem shows this route reaches the binary weight but ''not'' quantum phase, interference between non-commuting contexts, or the multi-outcome rule — those are inherited from the host, not derived.
* '''Penrose Objective Reduction''' is treated as the live ''discrimination target'' of the experiment, not as a rival to be ruled out by the framework and not as a co-contributing IOF mechanism. The earlier &quot;additive combined-rate&quot; reading, and a speculative ''Bridge Ansatz'' identifying ''E''&lt;sub&gt;G&lt;/sub&gt; with κ, have both been retired: the ansatz met its own failure criterion and survives only as explicitly speculative cosmogony in ''The Creation of Duality''. What remains is the clean division of variables — mass and geometry set ''τ''&lt;sub&gt;OR&lt;/sub&gt;; capacity and instability set the benchmark — and the recoverability classifier that keeps them apart.

== The measurement problem: where the Heisenberg cut sits ==

The framework gives an operational reading of the Heisenberg cut — the boundary between the quantum description used for the system and the classical description used for the apparatus and record. Standard interpretations place it variously: von Neumann showed it can be slid without changing predictions and treated its location as conventional; decoherence ties it to the rate of environmental coupling; objective-collapse proposals fix it universally at a mass scale.

IOF places the cut where the observer-apparatus's ''useful'' basis-tracking rate runs out relative to its basis-producing dynamics — at the locus where ''h''&lt;sub&gt;KS&lt;/sub&gt; = ''C''&lt;sub&gt;eff&lt;/sub&gt; ln 2. The operative quantity is the effective rate that genuinely constrains the reference, not the Landauer ceiling, which enters only as a thermodynamic upper limit. This is not an objective physical collapse boundary; it is a design-dependent threshold the experimenter can move — by throttling or widening the tracking loop at fixed temperature and power. It is observer-relative without being subjective: two loops on the same hardware place their cuts differently, but for a given configuration any inspector agrees where it sits.

That the cut ''moves'' is what the benchmark measures — improving the controller raises ''C''&lt;sub&gt;eff&lt;/sub&gt; and shifts the cut toward more chaotic basis-producing dynamics. This motion is expected control physics, not a departure from quantum mechanics: it is the operational signature of finite self-tracking, calibrated, and it is what the κ-benchmark reads off. The measurement problem took its sharpest form because the cut was treated as floating; the framework's narrower, testable claim is that for a given finite apparatus the cut is not floating but located, by the basis-tracking budget that apparatus devotes to its reference.

== Philosophical interpretation ==

''This section describes interpretive extensions beyond the empirical core. Nothing in it is load-bearing for the operational benchmark or the Penrose test. The interpretive positions themselves — Advaita Vedānta, relational quantum mechanics — do not stand or fall on an interferometry experiment; what is at stake is the framework's specific mapping into them. The framework's centre of gravity is the machinery, not the host ontology: swap the deterministic host for a pilot-wave or Everettian one and the machinery, and its exposure to experiment, are unchanged.''

The cleanest entry point to the interpretive position is ''[https://ignorantobserver.xyz/documents/Where_Did_the_Measurement_Basis_Come_From.pdf Where Did the Measurement Basis Come From?]'' (Dekker, 2026). It states the central move — the measurement basis as a physical variable with causal ancestry inside the same history as the system — addresses the standard objections, and names the position ''epistemically bounded ancestral correlation''.

A second interpretive piece, ''[https://ignorantobserver.xyz/documents/Response_to_Rovelli_on_the_Hard_Problem.pdf The Hard Problem Dissolved — But Into What?]'' (Dekker, 2026), engages Carlo Rovelli's Noema essay, marks the ground it shares with the framework, and identifies where the framework presses beyond Rovelli's deflationary physicalism toward a non-dual reading.

The interpretive layer is developed in dialogue with two existing positions. The first is Rovelli's relational quantum mechanics: the framework can be read as supplying a candidate physical mechanism — the ''C''&lt;sub&gt;eff&lt;/sub&gt; versus ''h''&lt;sub&gt;KS&lt;/sub&gt; inequality — for what makes an outcome relative to an observer rather than absolute. The second is the Advaita Vedānta tradition (Śaṅkara, Ramaṇa Maharṣi), in which the apparent independence of the experiencing subject from the perceived world is treated as a structural feature of ignorance (''avidyā'') rather than a metaphysical fact. The framework's σ&lt;sub&gt;θ&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;(''t'') — the growing self-opacity of an observer whose capacity cannot track its own apparatus — admits a structural analogy with avidyā as a physical limit on self-observation. The framework neither asserts that the analogy is more than structural nor that any experiment could confirm or refute Advaita as philosophy; it offers it as a way of locating the framework within a non-dual reading of measurement.

A separate IOF-internal derivation, ''[https://ignorantobserver.xyz/documents/The_Born_Rule_from_Finite_Observation.pdf The Born Rule from Finite Observation]'', obtains ''p''(θ) = cos²(θ/2) from finite-record geometry, with its scope fixed by the no-go companion ''[https://ignorantobserver.xyz/documents/Why_Fisher_Geometry_Gives_Binary_Born_but_Not_Quantum_Phase.pdf Why Fisher Geometry Gives Binary Born but Not Quantum Phase]''. Its metaphysical companion, ''[https://ignorantobserver.xyz/documents/Structural_Resonance.pdf Structural Resonance]'', documents how a structural reading of the ''Kaṭha Upaniṣad'' served as a disciplined search heuristic for the derivation, without claiming that Vedānta proves the Born rule.

A further speculative extension, ''[https://ignorantobserver.xyz/documents/The_Creation_of_Duality.pdf The Creation of Duality]'', asks whether space, time, objecthood, and gravity-like structure can themselves be read as features of a consistent finite-observer world-model. It retains the retired κ-to-gravity Bridge Ansatz only as explicitly speculative cosmogony, with no empirical warrant and no role in the experimental program above.

Readers who prefer to ignore the interpretive readings can evaluate the framework's empirical content from the [[#Technical proposal|Technical proposal]] and [[#The operational benchmark and the Penrose test|operational]] sections alone.

== What a clean result would, and would not, establish ==

* '''A clean benchmark''' (κ-scaling, double-exponential form, recoverable loss with ''R''&lt;sub&gt;rec&lt;/sub&gt; → 1) establishes that the framework's operational accounting of the observer's contribution is accurate and exhaustive in the tested regime — a validated instrument. It does '''not''' establish IOF over quantum mechanics, because the benchmark lies within quantum mechanics; that is the source of its credibility, not a limitation.
* '''The Penrose discrimination''' settles a question about gravity, not about IOF. An unrecoverable mass-geometry floor would vindicate Penrose and show quantum mechanics incomplete; its absence would constrain objective reduction at the tested scales. IOF supplies the calibrated subtrahend that makes either verdict trustworthy.
* '''The interpretation''' gains nothing a discriminating experiment could give it, and loses nothing a null could take away. It competes — by coherence and economy — with the other no-collapse readings; a Penrose null leaves it standing among them, neither crowned nor refuted.
* '''The avidyā mapping''' gains a concrete physical anchor for σ&lt;sub&gt;θ&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;(''t'') as a limit on self-observation, but the framework's claim there is structural, not metaphysical, and no interferometry result adjudicates the philosophical positions the mapping connects.

== Open review targets ==

These are the points on which the proposal should be attacked. The list is split into ''core'' targets, load-bearing for the operational claim, and ''further technical caveats''.

=== Core review targets ===

# '''Operationalising the recoverability classifier'''. Recoverability is the framework's ''position'', not an objection to it: the expected case is that the engineered loss is reference physics and returns under conditioning on a passive shadow log. The load-bearing question is whether ''R''&lt;sub&gt;rec&lt;/sub&gt; can be made a clean classifier in practice — whether the logging-fidelity budget can be quantified and frozen, and the recovered-versus-residual contrast separated, well enough to certify any surviving floor as not-the-observer's. If it cannot, the Penrose discrimination loses its subtrahend.
# '''Decoherence confound'''. Distinguishing ''V''&lt;sub&gt;IOF&lt;/sub&gt; from ''V''&lt;sub&gt;std&lt;/sub&gt; in practice is the central experimental challenge of the benchmark. A sufficiently flexible Lindblad / phase-noise model may absorb the predicted curve under suitable parameters; the benchmark gains independent force only when κ predicts visibility timing after thermal, readout, latency, pulse, actuator, and offline-recovery controls have been given every chance to win.
# '''Useful-capacity calibration'''. The framework relies on independent calibration of ''C''&lt;sub&gt;eff&lt;/sub&gt; as ''useful'' tracking capacity, not raw input power or the Landauer ceiling. Establishing that calibration empirically — via the Fisher-homogeneity module or an equivalent operational mapping — is the load-bearing engineering claim.
# '''Instability measure'''. Is ''h''&lt;sub&gt;KS&lt;/sub&gt; the right quantity for the basis-producing dynamics of real engineered apparatus? Many precision controllers are explicitly engineered to suppress chaos; their basis-defining degrees of freedom may show coloured noise or slow drift rather than positive-''h''&lt;sub&gt;KS&lt;/sub&gt; chaos in the Pesin sense. The protocol's expanding-dynamics gate exists to certify the regime before any κ-claim is read.
# '''Gaussian / independent-smearing assumption'''. The closed form ''V''&lt;sub&gt;obs&lt;/sub&gt; = ''V''&lt;sub&gt;std&lt;/sub&gt; · exp(−½ σ&lt;sub&gt;θ&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;) assumes σ&lt;sub&gt;θ&lt;/sub&gt; ≲ 1 rad, a Gaussian δθ, and independence between the basis-tracking and environmental channels. Non-Gaussian, heavy-tailed, or correlated errors would break the closed-form law and require a more general rate-distortion accounting.

=== Further technical caveats ===

# '''Rate-distortion extension to chaotic systems'''. The mapping from capacity ''C'' to tracking variance σ&lt;sub&gt;θ&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt; ≥ ''D''/(''C'' ln 2) assumes a high-rate coder, and the extension of the Data-Rate Theorem to nonlinear chaotic systems by substituting ''h''&lt;sub&gt;KS&lt;/sub&gt; is an explicit assumption, not a proven theorem.
# '''Prior-art and reparameterisation risk'''. The framework must show its predicted curve is not ordinary reference noise, phase jitter, or decoherence in new notation — the purpose of the adversarial-mimic analysis.
# '''Bell / locality consistency'''. The framework implies a non-conspiratorial violation of statistical measurement-independence (following Palmer 2024). A full consistency proof, including a no-signalling lemma for the hosted no-collapse embedding, has not been published.
# '''Scope of the Born-rule derivation'''. The companion note conditionally obtains the binary weight ''p''(θ) = cos²(θ/2) under two named premises (the Fisher capacity bridge and scalar-threshold homogeneity), both exposed through the Fisher-homogeneity module. Complex Hilbert space, tensor-product composition, unitary dynamics, the multi-outcome rule, and quantum phase are ''not'' derived — the no-go companion proves the route cannot reach them; they are inherited from the host structure.
# '''Peer-review status and replication'''. The framework has not undergone peer review and the experiments have not been performed. Its case must be evaluated on the documents and on the conduct of the prospective experiment, not on any external imprimatur.

== Invitation for review ==

This page is offered as a venue for substantive critique. The author is particularly interested in:

* '''From physicists in quantum control or precision interferometry''': can the recoverability classifier be implemented cleanly enough on real hardware to certify a residual floor as not-the-observer's, and what existing apparatus is best positioned to host the calibrated benchmark and the mass-geometry sweep?
* '''From decoherence theorists''': under what conditions does the double-exponential law overlap with compound-channel decoherence models in ways that would make a recoverable κ-channel and an unrecoverable one hard to separate in practice?
* '''From researchers in quantum foundations''': how should a non-conspiratorial, epistemically bounded violation of measurement-independence be evaluated against the superdeterminism / retrocausality / many-worlds landscape, and what would a satisfactory no-signalling consistency proof require?
* '''From researchers in information geometry''': is the Fisher capacity bridge the right identification of useful tracking capacity, is scalar-threshold homogeneity the natural reading of the BLQC threshold in a calibrated basis, and is Cencov-based selection the correct uniqueness route? Critique of the no-go boundary — what genuinely cannot be reached from finite-record geometry — is equally welcome.
* '''From philosophers of mind''': the Advaita / RQM interpretive layer is offered conditionally on the empirical core and as an interpretation among peers. Is the conditional structure presented clearly enough, or does it still amount to overreach?

Comments, references to prior or parallel work, and pointers to confounds or alternative explanations are all welcome. Substantive critique on the [[Talk:The Ignorant Observer Framework|talk page]] will be acknowledged in subsequent revisions.

== Documents ==

The framework's documents are published at [https://ignorantobserver.xyz ignorantobserver.xyz], grouped by role.

'''Foundational and bridges'''

* '''[https://ignorantobserver.xyz/documents/The_Ignorant_Observer.pdf The Ignorant Observer]''' — the foundational paper. The philosophical motivation (avidyā as a physical limit) and the technical groundwork from which the project grew.
* '''[https://ignorantobserver.xyz/documents/Where_Did_the_Measurement_Basis_Come_From.pdf Where Did the Measurement Basis Come From?]''' — the conceptual bridge for physicists. States the claim the framework makes about the measurement basis, addresses the standard objections, and names the position ''epistemically bounded ancestral correlation''.
* '''[https://ignorantobserver.xyz/documents/Bandwidth-Limited_Quantum_Control.pdf Bandwidth-Limited Quantum Control]''' — the operational layer: a finite-rate phase-reference control law and its operational benchmark (calibration arm, capacity and instability sweeps, shadow-log recoverability classifier, registered chaotic-corner module).
* '''[https://ignorantobserver.xyz/documents/A_Capacity-Calibrated_Protocol_for_Testing_Penrose_Objective_Reduction.pdf A Capacity-Calibrated Protocol for Testing Penrose Objective Reduction]''' — the prospective experiment: a mass-geometry sweep for an unrecoverable collapse floor, with the BLQC benchmark as its calibration prerequisite. Tests Penrose, not IOF.

'''Foundational extensions'''

* '''[https://ignorantobserver.xyz/documents/The_Born_Rule_from_Finite_Observation.pdf The Born Rule from Finite Observation]''' — derives the binary Born form ''p''(θ) = cos²(θ/2) from finite-record geometry, conditional on two named, empirically exposed assumptions. Does not derive complex Hilbert space, tensor products, unitary dynamics, phase, or the multi-outcome rule.
* '''[https://ignorantobserver.xyz/documents/Why_Fisher_Geometry_Gives_Binary_Born_but_Not_Quantum_Phase.pdf Why Fisher Geometry Gives Binary Born but Not Quantum Phase]''' — the no-go companion marking exactly where the finite-record route stops, so the boundary is a result rather than a gap.
* '''[https://ignorantobserver.xyz/documents/Structural_Resonance.pdf Structural Resonance]''' — a metaphysical companion explaining how a structural reading of the ''Kaṭha Upaniṣad'' served as a search heuristic for the derivation. Does not claim Vedānta proves the Born rule.

'''Supplements'''

* '''[https://ignorantobserver.xyz/documents/Forensic_Signatures.pdf Forensic Signatures]''' — retrospective screening of existing public data, reframed as classifying reference physics versus unrecoverable anomalies; the anomaly branch is registered, not asserted.
* '''[https://ignorantobserver.xyz/documents/The_Creation_of_Duality.pdf The Creation of Duality]''' — speculative extension on appearance, gravity, and information from self-ignorance; the κ-to-gravity ansatz is retained only as explicitly speculative cosmogony.
* '''[https://ignorantobserver.xyz/documents/The_Capacity-Backaction_Frontier.pdf The Capacity–Backaction Frontier]''' — application to cryogenic quantum error correction, comparing useful syndrome capacity against the instability induced by obtaining and using it.
* '''[https://ignorantobserver.xyz/documents/Biological_Observers.pdf Biological Observers]''' — exploratory supplement on biological timescales.

A full archival deposit is available on the Open Science Framework at [https://doi.org/10.17605/OSF.IO/FCDSN doi.org/10.17605/OSF.IO/FCDSN].

== References ==

* Bartlett, S. D., Rudolph, T., &amp; Spekkens, R. W. (2007). Reference frames, superselection rules, and quantum information. ''Reviews of Modern Physics'', 79(2), 555–609.
* Brukner, Č., &amp; Zeilinger, A. (1999). Operationally invariant information in quantum measurements. ''Physical Review Letters'', 83(17), 3354–3357.
* Huang, H.-Y., Kueng, R., &amp; Preskill, J. (2020). Predicting many properties of a quantum system from very few measurements. ''Nature Physics'', 16, 1050–1057.
* Nair, G. N., &amp; Evans, R. J. (2004). Stabilizability of stochastic linear systems with finite feedback data rates. ''SIAM Journal on Control and Optimization'', 43(2), 413–436.
* Palmer, T. (2024). Superdeterminism without conspiracy. ''Universe'', 10(1), 47.
* Penrose, R. (1996). On gravity's role in quantum state reduction. ''General Relativity and Gravitation'', 28(5), 581–600.
* Pusey, M. F., Barrett, J., &amp; Rudolph, T. (2012). On the reality of the quantum state. ''Nature Physics'', 8(6), 475–478.
* Rovelli, C. (1996). Relational quantum mechanics. ''International Journal of Theoretical Physics'', 35(8), 1637–1678.
* Tatikonda, S., &amp; Mitter, S. (2004). Control under communication constraints. ''IEEE Transactions on Automatic Control'', 49(7), 1056–1068.
* Weihs, G., Jennewein, T., Simon, C., Weinfurter, H., &amp; Zeilinger, A. (1998). Violation of Bell's inequality under strict Einstein locality conditions. ''Physical Review Letters'', 81(23), 5039–5043.
* Wootters, W. K. (1981). Statistical distance and Hilbert space. ''Physical Review D'', 23(2), 357–362.

== See also ==

* [[w:Quantum decoherence|Decoherence]] (Wikipedia)
* [[w:Relational quantum mechanics|Relational quantum mechanics]] (Wikipedia)
* [[w:Penrose interpretation|Penrose interpretation]] (Wikipedia)
* [[w:Data-rate theorem|Data-rate theorem]] (Wikipedia)

[[Category:Research projects]]
[[Category:Quantum mechanics]]
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  <page>
    <title>Banjo anatomy</title>
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      <text bytes="4596" sha1="oo54eswd6kgr0imff5ojrxj252ufhnf" xml:space="preserve">[[Banjo]] anatomy refers to the structural and functional components used in creating a banjo. A banjo can first be seen split into two sections: the neck and the pot assembly.&lt;ref&gt;https://banjo.com/anatomy-and-design-understanding-the-construction-of-the-banjo/&lt;/ref&gt;
== The Neck and its parts ==
The neck of the banjo is a long piece of wood there to give distance between the nut and the bridge, which is known as the [[w:Scale_length_(string_instruments) | scale length]]. The scale length is the distance of the string that vibrates when it is played.

The neck is usually made out of a dense wood such as Maple, Mahogany, or Walnut.&lt;ref&gt;https://banjonews.com/2012-02/does_neck_wood_matter.html&lt;/ref&gt; It was originally made out of a single piece of wood, and as manufacturing advanced, manufacturers constructed them from multiple pieces or used lamination to improve durability. 

=== Headstock ===
On a banjo, the headstock contains four tuning machines and may include a manufacturer logo or decorative inlays and shaping.

The tuning machine is made of three parts: pegs, posts, and gears.

=== Nut ===
The nut is a small piece of bone, plastic, or metal located between the headstock and fingerboard. It spaces the strings evenly and establishes one end of the instrument's vibrating string length. This identifies one end of a strings scale length to play a certain note, unless a string gets pressed down on the fingerboard. The other end of the strings scale length is located at the bridge.

=== Fingerboard ===
The fingerboard on the most common banjos refers to the flat surface attached to the neck. It contains frets or fret markers and provides the area where the player presses the strings to change pitch. People often create banjos with modifications that use an unfretted fingerboard or radiused fingerboard.

=== Heel ===
The heel refers to the widened portion of the neck that attached to the pot. The way this fits affects neck alignment. See Neck Attachment Systems for more on how this section attaches.

== The Pot Assembly and its parts ==
==== Rim ====
The rim is a wooden hollowed out cylinder that serves as the foundation for other parts to attach. These are often made from multiple layers of steamed and bent wood, but can also be made as block rims that uses multiple solid blocks glued together.
==== Head ====
The head is the tensioned thin material that works with the vibrations from the strings transfer to the bridge, and then to the head, vibrating across the diameter. On a banjo with a resonator, the sound also projects out through the head more than the harder material of the resonator. Most heads are made out of animal skins or mylar.
=== Neck Attachment System ===
==== Coordinator rod ====

==== Dowel stick ====

==== Glue and Bolts ====

=== Head Tension System ===
==== Tacked Heads ====

==== Hooks and Nuts ====

==== Bracket Shoes ====

==== Flange ====

==== Tension Hoop ====


==== Tailpiece ====
The tailpiece holds and spaces out the strings at the lower end of the banjo. It can have an adjustable break angle, which can influence tone.&lt;ref&gt;{{Cite web|url=https://banjonews.com/2010-01/string_break_angle.html|title=Banjo Newsletter|website=banjonews.com|access-date=2026-06-12}}&lt;/ref&gt;
==== Bridge ====
The bridge is an independent wooden piece that acts as one side of the scale length, that vibrates to the head and produces more volume and sustain. Bridges vary by the type of wood, mass, and contact points with the head.
==== Tone Ring ====
A tone ring can be made from metal (Steel, Rolled Brass, Bell Bronze, and other alloys), wood, or carbon-fiber. The general rule for the tone is that the harder and stiffer the material, the more bright the sound, and softer materials provide a warmer tone. Increased weight in the tone ring also increases the sustain. &lt;ref&gt;{{Cite web|url=https://blog.deeringbanjos.com/what-banjo-tone-rings-do|title=What Does a Banjo Tone Ring Do?|last=Hunn|first=Barry|website=blog.deeringbanjos.com|language=en-us|access-date=2026-06-04}}&lt;/ref&gt;
==== Resonator ====
A resonator refers to the wooden back that is attached to the pot. This allows for more sound to project through the head of the banjo, rather than the sound that already projects towards your abdomen.
==== Armrest ====
An armrest is a metal or wooden piece attached to the rim, used above the head of the banjo, so that the wrist and forearm don't rest directly onto the head of the banjo. This increases comfort and decreases the sound loss from resting an arm on the resonating head of a banjo.

== Sources ==

[[Category:Banjo]]</text>
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== Licensing ==
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    <title>Wikimedia concerns with European copyright rules including AI and scientific research</title>
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      <comment>Zagorski is Policy Dir</comment>
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      <text bytes="5987" sha1="krob57syu291j2gp2kha7sm0n55ztsw" xml:space="preserve">:''This discusses a 2026-06-18 interview with Dimitri Zagorski&lt;ref name=Dimi&gt;&lt;!--Dimitar Zagorski, 3rd--&gt;{{cite Q|Q130719781}}&lt;/ref&gt; about the European Commission's targeted public consultation on copyright law, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-06-27 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
&lt;!--[[File:Wikimedia concerns with European copyright rules including AI and scientific research.webm|thumb|2026-06-18 interview with Dimi Zagorski about Wikimedia concerns with European copyright rules including AI and scientific research.]]--&gt; 
&lt;!--[[File:Wikimedia concerns with European copyright rules including AI and scientific research.ogg|thumb|29:00 mm:ss excerpts from a 2026-06-18 interview with Dimi Zagorski about Wikimedia concerns with European copyright rules including AI and scientific research.]]--&gt; 

Dimitri (&quot;Dimi&quot;) Zagorski&lt;ref name=Dimi/&gt; discusses the European Commission's targeted public consultation on copyright law and other issues that concern the Wikimedia Foundation. Zagorski is Policy Director for Wikimedia Europe in Brussels. He is interviewed by Spencer Graves.&lt;ref&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt;

As background for this interview, we review the structure of the government of the [[w:European Union|European Union]] (EU).

== European Union ==

The European Union (EU) is a political and economic union of 27 member states located primarily in Europe with [[w:Special territories of members of the European Economic Area|32 special territories]] or subnational units of EU member states stretching from [[w:Greenland|Greenland]] to the southern Indian ocean and the South Pacific. The EU has a population of over 450 million and nominal gross domestic product (GDP) of around €19 trillion in 2025; this makes it roughly one sixth of the global economy. The [[w:Institutions of the European Union|Institutions of the European Union]] encompass seven principal decision-making bodies:

# [[w:European Parliament|European Parliament]], whose approval is required for proposed legislation to become law. It currently has 720 members (MEPs). It functions roughly like the [[w:United States House of Representatives|House of Representatives in the US]], in that MEPs can amend or reject proposed legislation, though they cannot initiate legislation. 
# [[w:European Council|European Council]] of heads of state or government.
# [[w:Council of the European Union|Council of the European Union]], often referred to simply as the Council and less formally as the &quot;Council of Ministers&quot;. This is a legislative body, which works with the European Parliament to amend and approve or veto proposals of the European Commission, which holds the right of initiative; neither the Council nor the Parliament can initiate legislation. The Presidency of the council is not a single post, but is held by a member state's government and rotates every six months. It functions roughly like the [[w:United States Senate|Senate in the US]] but cannot initiate legislation. 
# [[w:European Commission|European Commission]] (EC), the executive cabinet of the European Union. Currently, there is one Commissioner per member state, including the president (currently [[w:Ursula von der Leyen|Ursula von der Leyen]]), but members are bound by their oath of office to represent the interest of the EU as a whole rather than their home state. All EU legislation must be initiated by the Commission, to be amended and approved or vetoed by the European Parliament and the Council. 
# [[w:Court of Justice of the European Union|Court of Justice of the European Union]].
# [[w:European Central Bank|European Central Bank]]. 
# [[w:European Court of Auditors|European Court of Auditors]]. 

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

&lt;!--== Bibliography ==--&gt;
[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Politics]]
[[Category:Media literacy]]
[[Category:Wikimedia]]
[[Category:Wikimedia Foundation]]
[[Category:Wikimedia Foundation staff]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
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    <revision>
      <id>2815423</id>
      <parentid>2815422</parentid>
      <timestamp>2026-06-12T16:58:52Z</timestamp>
      <contributor>
        <username>DavidMCEddy</username>
        <id>218607</id>
      </contributor>
      <minor />
      <comment>typo</comment>
      <origin>2815423</origin>
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      <format>text/x-wiki</format>
      <text bytes="5987" sha1="c1wgx8tl96mk2vd93inhceaj0qgpssj" xml:space="preserve">:''This discusses a 2026-06-18 interview with Dimitar Zagorski&lt;ref name=Dimi&gt;&lt;!--Dimitar Zagorski, 3rd--&gt;{{cite Q|Q130719781}}&lt;/ref&gt; about the European Commission's targeted public consultation on copyright law, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-06-27 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
&lt;!--[[File:Wikimedia concerns with European copyright rules including AI and scientific research.webm|thumb|2026-06-18 interview with Dimi Zagorski about Wikimedia concerns with European copyright rules including AI and scientific research.]]--&gt; 
&lt;!--[[File:Wikimedia concerns with European copyright rules including AI and scientific research.ogg|thumb|29:00 mm:ss excerpts from a 2026-06-18 interview with Dimi Zagorski about Wikimedia concerns with European copyright rules including AI and scientific research.]]--&gt; 

Dimitar (&quot;Dimi&quot;) Zagorski&lt;ref name=Dimi/&gt; discusses the European Commission's targeted public consultation on copyright law and other issues that concern the Wikimedia Foundation. Zagorski is Policy Director for Wikimedia Europe in Brussels. He is interviewed by Spencer Graves.&lt;ref&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt;

As background for this interview, we review the structure of the government of the [[w:European Union|European Union]] (EU).

== European Union ==

The European Union (EU) is a political and economic union of 27 member states located primarily in Europe with [[w:Special territories of members of the European Economic Area|32 special territories]] or subnational units of EU member states stretching from [[w:Greenland|Greenland]] to the southern Indian ocean and the South Pacific. The EU has a population of over 450 million and nominal gross domestic product (GDP) of around €19 trillion in 2025; this makes it roughly one sixth of the global economy. The [[w:Institutions of the European Union|Institutions of the European Union]] encompass seven principal decision-making bodies:

# [[w:European Parliament|European Parliament]], whose approval is required for proposed legislation to become law. It currently has 720 members (MEPs). It functions roughly like the [[w:United States House of Representatives|House of Representatives in the US]], in that MEPs can amend or reject proposed legislation, though they cannot initiate legislation. 
# [[w:European Council|European Council]] of heads of state or government.
# [[w:Council of the European Union|Council of the European Union]], often referred to simply as the Council and less formally as the &quot;Council of Ministers&quot;. This is a legislative body, which works with the European Parliament to amend and approve or veto proposals of the European Commission, which holds the right of initiative; neither the Council nor the Parliament can initiate legislation. The Presidency of the council is not a single post, but is held by a member state's government and rotates every six months. It functions roughly like the [[w:United States Senate|Senate in the US]] but cannot initiate legislation. 
# [[w:European Commission|European Commission]] (EC), the executive cabinet of the European Union. Currently, there is one Commissioner per member state, including the president (currently [[w:Ursula von der Leyen|Ursula von der Leyen]]), but members are bound by their oath of office to represent the interest of the EU as a whole rather than their home state. All EU legislation must be initiated by the Commission, to be amended and approved or vetoed by the European Parliament and the Council. 
# [[w:Court of Justice of the European Union|Court of Justice of the European Union]].
# [[w:European Central Bank|European Central Bank]]. 
# [[w:European Court of Auditors|European Court of Auditors]]. 

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

&lt;!--== Bibliography ==--&gt;
[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Politics]]
[[Category:Media literacy]]
[[Category:Wikimedia]]
[[Category:Wikimedia Foundation]]
[[Category:Wikimedia Foundation staff]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
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        <username>DavidMCEddy</username>
        <id>218607</id>
      </contributor>
      <comment>add mention of the statuatory review of the 2019 Directive on Copyright in the Digital Single Market</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6140" sha1="htkis1qyzm4jjiiqlbaga17hdpojaqz" xml:space="preserve">:''This discusses a 2026-06-18 interview with Dimitar Zagorski&lt;ref name=Dimi&gt;&lt;!--Dimitar Zagorski, 3rd--&gt;{{cite Q|Q130719781}}&lt;/ref&gt; about the European Commission's targeted public consultation on copyright law, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-06-27 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
&lt;!--[[File:Wikimedia concerns with European copyright rules including AI and scientific research.webm|thumb|2026-06-18 interview with Dimi Zagorski about Wikimedia concerns with European copyright rules including AI and scientific research.]]--&gt; 
&lt;!--[[File:Wikimedia concerns with European copyright rules including AI and scientific research.ogg|thumb|29:00 mm:ss excerpts from a 2026-06-18 interview with Dimi Zagorski about Wikimedia concerns with European copyright rules including AI and scientific research.]]--&gt; 

Dimitri (&quot;Dimi&quot;) Zagorski&lt;ref name=Dimi/&gt; discusses the European Commission's targeted public consultation on copyright law and other issues that concern the Wikimedia Foundation including the statutory review of the 2019 [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Digital Single Market]]. Zagorski is Policy Director for Wikimedia Europe in Brussels. He is interviewed by Spencer Graves.&lt;ref&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt;

As background for this interview, we review the structure of the government of the [[w:European Union|European Union]] (EU).

== European Union ==

The European Union (EU) is a political and economic union of 27 member states located primarily in Europe with [[w:Special territories of members of the European Economic Area|32 special territories]] or subnational units of EU member states stretching from [[w:Greenland|Greenland]] to the southern Indian ocean and the South Pacific. The EU has a population of over 450 million and nominal gross domestic product (GDP) of around €19 trillion in 2025; this makes it roughly one sixth of the global economy. The [[w:Institutions of the European Union|Institutions of the European Union]] encompass seven principal decision-making bodies:

# [[w:European Parliament|European Parliament]], whose approval is required for proposed legislation to become law. It currently has 720 members (MEPs). It functions roughly like the [[w:United States House of Representatives|House of Representatives in the US]], in that MEPs can amend or reject proposed legislation, though they cannot initiate legislation. 
# [[w:European Council|European Council]] of heads of state or government.
# [[w:Council of the European Union|Council of the European Union]], often referred to simply as the Council and less formally as the &quot;Council of Ministers&quot;. This is a legislative body, which works with the European Parliament to amend and approve or veto proposals of the European Commission, which holds the right of initiative; neither the Council nor the Parliament can initiate legislation. The Presidency of the council is not a single post, but is held by a member state's government and rotates every six months. It functions roughly like the [[w:United States Senate|Senate in the US]] but cannot initiate legislation. 
# [[w:European Commission|European Commission]] (EC), the executive cabinet of the European Union. Currently, there is one Commissioner per member state, including the president (currently [[w:Ursula von der Leyen|Ursula von der Leyen]]), but members are bound by their oath of office to represent the interest of the EU as a whole rather than their home state. All EU legislation must be initiated by the Commission, to be amended and approved or vetoed by the European Parliament and the Council. 
# [[w:Court of Justice of the European Union|Court of Justice of the European Union]].
# [[w:European Central Bank|European Central Bank]]. 
# [[w:European Court of Auditors|European Court of Auditors]]. 

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

&lt;!--== Bibliography ==--&gt;
[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Politics]]
[[Category:Media literacy]]
[[Category:Wikimedia]]
[[Category:Wikimedia Foundation]]
[[Category:Wikimedia Foundation staff]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
      <sha1>htkis1qyzm4jjiiqlbaga17hdpojaqz</sha1>
    </revision>
    <revision>
      <id>2815427</id>
      <parentid>2815426</parentid>
      <timestamp>2026-06-12T17:45:17Z</timestamp>
      <contributor>
        <username>DavidMCEddy</username>
        <id>218607</id>
      </contributor>
      <comment>wdsmth</comment>
      <origin>2815427</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6176" sha1="h6vhbfv6350uak4m9wtq6o6gka4wit7" xml:space="preserve">:''This discusses a 2026-06-18 interview with Dimitar Zagorski&lt;ref name=Dimi&gt;&lt;!--Dimitar Zagorski, 3rd--&gt;{{cite Q|Q130719781}}&lt;/ref&gt; about the European Commission's targeted public consultation on copyright law, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-06-27 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
&lt;!--[[File:Wikimedia concerns with European copyright rules including AI and scientific research.webm|thumb|2026-06-18 interview with Dimi Zagorski about Wikimedia concerns with European copyright rules including AI and scientific research.]]--&gt; 
&lt;!--[[File:Wikimedia concerns with European copyright rules including AI and scientific research.ogg|thumb|29:00 mm:ss excerpts from a 2026-06-18 interview with Dimi Zagorski about Wikimedia concerns with European copyright rules including AI and scientific research.]]--&gt; 

Dimitri (&quot;Dimi&quot;) Zagorski, 3rd.,&lt;ref name=Dimi/&gt; discusses the European Commission's targeted public consultation on copyright law and other issues that concern the Wikimedia Foundation including AI, scientific research, and the statutory review of the 2019 [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Digital Single Market]]. Zagorski is Policy Director for Wikimedia Europe in Brussels. He is interviewed by Spencer Graves.&lt;ref&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt;

As background for this interview, we review the structure of the government of the [[w:European Union|European Union]] (EU).

== European Union ==

The European Union (EU) is a political and economic union of 27 member states located primarily in Europe with [[w:Special territories of members of the European Economic Area|32 special territories]] or subnational units of EU member states stretching from [[w:Greenland|Greenland]] to the southern Indian ocean and the South Pacific. The EU has a population of over 450 million and nominal gross domestic product (GDP) of around €19 trillion in 2025; this makes it roughly one sixth of the global economy. The [[w:Institutions of the European Union|Institutions of the European Union]] encompass seven principal decision-making bodies:

# [[w:European Parliament|European Parliament]], whose approval is required for proposed legislation to become law. It currently has 720 members (MEPs). It functions roughly like the [[w:United States House of Representatives|House of Representatives in the US]], in that MEPs can amend or reject proposed legislation, though they cannot initiate legislation. 
# [[w:European Council|European Council]] of heads of state or government.
# [[w:Council of the European Union|Council of the European Union]], often referred to simply as the Council and less formally as the &quot;Council of Ministers&quot;. This is a legislative body, which works with the European Parliament to amend and approve or veto proposals of the European Commission, which holds the right of initiative; neither the Council nor the Parliament can initiate legislation. The Presidency of the council is not a single post, but is held by a member state's government and rotates every six months. It functions roughly like the [[w:United States Senate|Senate in the US]] but cannot initiate legislation. 
# [[w:European Commission|European Commission]] (EC), the executive cabinet of the European Union. Currently, there is one Commissioner per member state, including the president (currently [[w:Ursula von der Leyen|Ursula von der Leyen]]), but members are bound by their oath of office to represent the interest of the EU as a whole rather than their home state. All EU legislation must be initiated by the Commission, to be amended and approved or vetoed by the European Parliament and the Council. 
# [[w:Court of Justice of the European Union|Court of Justice of the European Union]].
# [[w:European Central Bank|European Central Bank]]. 
# [[w:European Court of Auditors|European Court of Auditors]]. 

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

&lt;!--== Bibliography ==--&gt;
[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Politics]]
[[Category:Media literacy]]
[[Category:Wikimedia]]
[[Category:Wikimedia Foundation]]
[[Category:Wikimedia Foundation staff]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
      <sha1>h6vhbfv6350uak4m9wtq6o6gka4wit7</sha1>
    </revision>
  </page>
  <page>
    <title>User:Juandev/Userviews 2015-2026</title>
    <ns>2</ns>
    <id>330157</id>
    <revision>
      <id>2815416</id>
      <parentid>2815369</parentid>
      <timestamp>2026-06-12T16:02:51Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>+2 tabs</comment>
      <origin>2815416</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5137" sha1="0cgr7i6f4rkq8lyruwfbjq6khlnmf43" xml:space="preserve">Views by users (not bots)
{| class=&quot;wikitable sortable&quot;
!Project
!User
!Pages created
!Total views
!Daily avarage views
!Total siteviews
!Notes
|-
|en.wv
|Juandev
|97
|94,702
|24
|447,002,321
|
|-
|cs.wv
|Juandev
|1317
|588,665
|147
|10,885,049
|
|-
|cs.wv
|Mmh
|460
|609,433
|152
|10,885,049
|
|-
|cs.wv
|Kychot
|2163
|1,059,364
|265
|10,885,049
|
|-
|cs.wv
|Kusurija
|401
|247,470
|62
|10,885,049
|
|-
|cs.wv
|Dan Polansky
|3
|1,428
|0
|10,885,049
|
|-
|cs.wv
|Danny B.
|3
|858
|0
|10,885,049
|
|-
|cs.wv
|Milda
|4
|4,775
|1
|10,885,049
|
|-
|cs.wv
|JAn Dudík
|3
|3,830
|1
|10,885,049
|
|-
|cs.wv
|Lenka64
|12
|65,139
|16
|10,885,049
|
|-
|cs.wp
|Juandev
|1,425
|30,242,025
|7,562
|9,121,647,039
|
|}

== Top 5 in 2025 ==

=== Top 5 of en.wv ===
{| class=&quot;wikitable&quot;
!No.
!Name of a resource
!Subject
!Type
!Age group
!Page status
!Views
!Project main page
!Project completition status
!Notes
|-
|1
|[[10 Principles of Economics]]
|economy
|text
|higher
|complete
|124,683
|same
|100 %
|
|-
|2
|[[Hello, world!]]
|programming
|data collection
|all
|ongoing
|86,413
|same
|80 %
|Programe scripts data collection
|-
|3
|[[Alphabet/Spanish alphabet]]
|linguistics
|data collection/text
|all
|complete
|73,161
|[[Alphabet]]
|10 %
|
|-
|4
|[[WikiJournal of Science/Earth-grazing meteoroid of 13 October 1990]]
|astronomy
|reviewed article
|higher
|complete
|54,317
|same
|100 %
|
|-
|5
|[[Database Examples/Northwind]]
|programming
|data collection/text
|higher
|complete
|52,732
|Database Examples
|
|
|}

=== Top 5 by Juandev on en.wv ===
{| class=&quot;wikitable&quot;
!No.
!Name of the resource
!Subject
!Type
!Age group
!Page status
!Viewes
!Project main page
!Project completition status
!Notes
|-
|1
|[[Solar System, interactive/Earth/Solar System overview]]
|astronomy
|interactive
|10-13
|completed
|1,405
|[[Solar System, interactive]]
|90 %
|
|-
|2
|[[Fairy Rings]]
|mycology
|data collection
|all
|ongoing
|1,322
|same
|10 %
|
|-
|3
|[[Solar System, interactive]]
|astronomy
|interactive
|10-13
|completed
|1,112
|same
|90 %
|
|-
|4
|[[Solar System, interactive/Earth/Solar System overview/Earth]]
|astronomy
|interactive
|10-13
|completed
|993
|[[Solar System, interactive]]
|90 %
|
|-
|5
|[[Solar System, interactive/Earth/Solar System overview/Mars]]
|astronomy
|interactive
|10-13
|completed
|800
|[[Solar System, interactive]]
|90 %
|
|}

=== Top 5 of cs.wv ===
{| class=&quot;wikitable&quot;
!No.
!Name of the resource
!Subject
!Type
!Age group
!Page status
!Viewes
!Project main page
!Project completition status
!Notes
|-
|1
|[[:cs:ClamTk|ClamTk]]
|IT, software usage
|disambiguation
|higher
|complete
|16,697
|same
|100 %
|Juandev
|-
|2
|[[:cs:Nářeční_výrazy_v_češtině|Nářeční výrazy v češtině]]
|linguistics
|data collection/text
|higher
|ongoing
|12,989
|same
|50 %
|
|-
|3
|[[:cs:DokuWiki|DokuWiki]]
|IT, software usage
|disambiguation, text
|higher
|ongoing
|3,677
|same
|10 %
|Kychot
|-
|4
|[[:cs:Latina_pro_začátečníky/Lekce_1|Latina pro začátečníky/Lekce 1]]
|linguistics
|lecture/text
|higher
|complete
|3,403
|[[:cs:Latina_pro_začátečníky|Latina pro začátečníky]]
|100 %
|Mmh
|-
|5
|[[:cs:Němčina/Základní_slovní_zásoba/Barvy|Němčina/Základní slovní zásoba/Barvy]]
|linguistics
|data collection/text
|higher
|complete
|3,350
|[[:cs:Němčina|Němčina]]
|10 %
|Juandev, Mmh
|}

Top 5 by Juandev on cs.wv
{| class=&quot;wikitable&quot;
!No.
!Name of a resource
!Subject
!Type
!Age group
!Page status
!Project main page
!Project completition statu
!Total 10 year views
!Notes
|-
|1
|[[Solar System, interactive/Earth/Solar System overview]]
|astronomy
|interactive
|10-13
|completed
|[[Solar System, interactive]]
|90 %
|10,168
|
|-
|2
|[[Solar System, interactive]]
|astronomy
|interactive
|10-13
|completed
|same
|90 %
|4,568
|
|-
|3
|[[Fairy Rings]]
|mycology
|data collection
|all
|ongoing
|same
|10 %
|3,663
|
|-
|4
|[[Solar System, interactive/Earth/Solar System overview/Earth]]
|astronomy
|interactive
|10-13
|completed
|[[Solar System, interactive]]
|90 %
|2,703
|
|-
|5
|[[Solar System, interactive/Earth/Solar System overview/Mars]]
|astronomy
|interactive
|10-13
|completed
|[[Solar System, interactive]]
|90 %
|2,258
|
|-
|6
|[[Spanish/Bilingual Spanish-English Dictionary]]
|linguistics
|main page
|all
|not completed
|same
|0
|2,228
|
|-
|7
|[[System/Earth/Solar System overview/Sun/Sun inside]]
|astronomy
|interactive
|10-13
|completed
|[[Solar System, interactive]]
|90 %
|1,943
|
|-
|8
|[[Plant tissue culture/Lab]]
|botany
|
|
|
|
|
|1,722
|
|-
|9
|[[BCP/Salix]]
|botany
|
|
|
|
|
|1,669
|
|-
|10
|[[Solar System, interactive/Earth]]
|astronomy
|interactive
|10-13
|completed
|[[Solar System, interactive]]
|90 %
|1,525
|
|}

Němčina/Základní slovní zásoba/Barvy; Klávesové zkratky pro zvláštní znaky; ClamTk; Ogg Theora/Zpracování videí pro Wikimedia Commons ve Windows; Práce na Wikipedii; Karma (ohřívač)/Mora PO 35/Juandev/Časté výbuchy a zhasínání; Wiki Rádio; Kontrola pravopisu; Projekt: Kvetení rostlin; 

== Czech Wikipedia ==
{| class=&quot;wikitable&quot;
!User
!Pages
!Total pageviews
!Daily average
!Notes
|-
|Juandev
|1,425
|30,242,025
|7,562
|
|}</text>
      <sha1>0cgr7i6f4rkq8lyruwfbjq6khlnmf43</sha1>
    </revision>
    <revision>
      <id>2815419</id>
      <parentid>2815416</parentid>
      <timestamp>2026-06-12T16:23:55Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>clean up</comment>
      <origin>2815419</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4727" sha1="jryj59o6gktsh0pdymq6k0da980biyl" xml:space="preserve">== Views by users (not bots) on various projects ==
{| class=&quot;wikitable sortable&quot;
!Project
!User
!Pages created
!Total views
!Daily avarage views
!Total siteviews
!Notes
|-
|en.wv
|Juandev
|97
|94,702
|24
|447,002,321
|
|-
|cs.wv
|Juandev
|1317
|588,665
|147
|10,885,049
|
|-
|cs.wv
|Mmh
|460
|609,433
|152
|10,885,049
|
|-
|cs.wv
|Kychot
|2163
|1,059,364
|265
|10,885,049
|
|-
|cs.wv
|Kusurija
|401
|247,470
|62
|10,885,049
|
|-
|cs.wv
|Dan Polansky
|3
|1,428
|0
|10,885,049
|
|-
|cs.wv
|Danny B.
|3
|858
|0
|10,885,049
|
|-
|cs.wv
|Milda
|4
|4,775
|1
|10,885,049
|
|-
|cs.wv
|JAn Dudík
|3
|3,830
|1
|10,885,049
|
|-
|cs.wv
|Lenka64
|12
|65,139
|16
|10,885,049
|
|-
|cs.wp
|Juandev
|1,425
|30,242,025
|7,562
|9,121,647,039
|
|}

== Top 5 in 2025 ==

=== Top 5 of en.wv ===
{| class=&quot;wikitable&quot;
!No.
!Name of a resource
!Subject
!Type
!Age group
!Page status
!Views
!Project main page
!Project completition status
!Notes
|-
|1
|[[10 Principles of Economics]]
|economy
|text
|higher
|complete
|124,683
|same
|100 %
|
|-
|2
|[[Hello, world!]]
|programming
|data collection
|all
|ongoing
|86,413
|same
|80 %
|Programe scripts data collection
|-
|3
|[[Alphabet/Spanish alphabet]]
|linguistics
|data collection/text
|all
|complete
|73,161
|[[Alphabet]]
|10 %
|
|-
|4
|[[WikiJournal of Science/Earth-grazing meteoroid of 13 October 1990]]
|astronomy
|reviewed article
|higher
|complete
|54,317
|same
|100 %
|
|-
|5
|[[Database Examples/Northwind]]
|programming
|data collection/text
|higher
|complete
|52,732
|Database Examples
|
|
|}

=== Top 5 by Juandev on en.wv ===
{| class=&quot;wikitable&quot;
!No.
!Name of the resource
!Subject
!Type
!Age group
!Page status
!Viewes
!Project main page
!Project completition status
!Notes
|-
|1
|[[Solar System, interactive/Earth/Solar System overview]]
|astronomy
|interactive
|10-13
|completed
|1,405
|[[Solar System, interactive]]
|90 %
|
|-
|2
|[[Fairy Rings]]
|mycology
|data collection
|all
|ongoing
|1,322
|same
|10 %
|
|-
|3
|[[Solar System, interactive]]
|astronomy
|interactive
|10-13
|completed
|1,112
|same
|90 %
|
|-
|4
|[[Solar System, interactive/Earth/Solar System overview/Earth]]
|astronomy
|interactive
|10-13
|completed
|993
|[[Solar System, interactive]]
|90 %
|
|-
|5
|[[Solar System, interactive/Earth/Solar System overview/Mars]]
|astronomy
|interactive
|10-13
|completed
|800
|[[Solar System, interactive]]
|90 %
|
|}

=== Top 5 of cs.wv ===
{| class=&quot;wikitable&quot;
!No.
!Name of the resource
!Subject
!Type
!Age group
!Page status
!Viewes
!Project main page
!Project completition status
!Notes
|-
|1
|[[:cs:ClamTk|ClamTk]]
|IT, software usage
|disambiguation
|higher
|complete
|16,697
|same
|100 %
|Juandev
|-
|2
|[[:cs:Nářeční_výrazy_v_češtině|Nářeční výrazy v češtině]]
|linguistics
|data collection/text
|higher
|ongoing
|12,989
|same
|50 %
|
|-
|3
|[[:cs:DokuWiki|DokuWiki]]
|IT, software usage
|disambiguation, text
|higher
|ongoing
|3,677
|same
|10 %
|Kychot
|-
|4
|[[:cs:Latina_pro_začátečníky/Lekce_1|Latina pro začátečníky/Lekce 1]]
|linguistics
|lecture/text
|higher
|complete
|3,403
|[[:cs:Latina_pro_začátečníky|Latina pro začátečníky]]
|100 %
|Mmh
|-
|5
|[[:cs:Němčina/Základní_slovní_zásoba/Barvy|Němčina/Základní slovní zásoba/Barvy]]
|linguistics
|data collection/text
|higher
|complete
|3,350
|[[:cs:Němčina|Němčina]]
|10 %
|Juandev, Mmh
|}

== Juandev 10 year total views on en.wv ==
{| class=&quot;wikitable&quot;
!No.
!Name of a resource
!Subject
!Type
!Age group
!Page status
!Project main page
!Project completition statu
!Total 10 year views
!Notes
|-
|1
|[[Solar System, interactive/Earth/Solar System overview]]
|astronomy
|interactive
|10-13
|completed
|[[Solar System, interactive]]
|90 %
|10,168
|
|-
|2
|[[Solar System, interactive]]
|astronomy
|interactive
|10-13
|completed
|same
|90 %
|4,568
|
|-
|3
|[[Fairy Rings]]
|mycology
|data collection
|all
|ongoing
|same
|10 %
|3,663
|
|-
|4
|[[Solar System, interactive/Earth/Solar System overview/Earth]]
|astronomy
|interactive
|10-13
|completed
|[[Solar System, interactive]]
|90 %
|2,703
|
|-
|5
|[[Solar System, interactive/Earth/Solar System overview/Mars]]
|astronomy
|interactive
|10-13
|completed
|[[Solar System, interactive]]
|90 %
|2,258
|
|-
|6
|[[Spanish/Bilingual Spanish-English Dictionary]]
|linguistics
|main page
|all
|not completed
|same
|0
|2,228
|
|-
|7
|[[System/Earth/Solar System overview/Sun/Sun inside]]
|astronomy
|interactive
|10-13
|completed
|[[Solar System, interactive]]
|90 %
|1,943
|
|-
|8
|[[Plant tissue culture/Lab]]
|botany
|
|
|
|
|
|1,722
|
|-
|9
|[[BCP/Salix]]
|botany
|
|
|
|
|
|1,669
|
|-
|10
|[[Solar System, interactive/Earth]]
|astronomy
|interactive
|10-13
|completed
|[[Solar System, interactive]]
|90 %
|1,525
|
|}</text>
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The goal of this Wikiversity course is to provide learning materials for pre-university mathematics.

===[[/Glossary/|Glossary]]===

===Algebra===
*[[/Algebra Functions/|Functions]]

===Geometry===

===Statistics===</text>
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